ISSUE 2005 PROGRESS IN PHYSICS
VOLUME 1
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Chief Editor D. Rabounski A New Method to Measure the Speed of Gravitation...... 3 Dmitri Rabounski Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes. . . 7 [email protected] F. Smarandache Associate Editors F. Smarandache A New Form of Matter — Unmatter, Composed of Particles Prof. Florentin Smarandache and Anti-Particles...... 9 [email protected] L. Nottale Fractality Field in the Theory of Scale Relativity...... 12 Dr. Larissa Borissova [email protected] L. Borissova, D. Rabounski On the Possibility of Instant Displacements in Stephen J. Crothers the Space-Time of General Relativity...... 17 [email protected] C. Castro On Dual Phase-Space Relativity, the Machian Principle and Mod- Department of Mathematics, University of ified Newtonian Dynamics...... 20 New Mexico, 200 College Road, Gallup, NM 87301, USA C. Castro, M. Pavsiˇ cˇ The Extended Relativity Theory in Clifford Spaces...... 31 K. Dombrowski Rational Numbers Distribution and Resonance...... 65 Copyright c Progress in Physics, 2005 S. J. Crothers On the General Solution to Einstein’s Vacuum Field and Its All rights reserved. Any part of Progress in Implications for Relativistic Degeneracy...... 68 Physics howsoever used in other publica- tions must include an appropriate citation S. J. Crothers On the Ramifications of the Schwarzschild Space-Time Metric..74 of this journal. S. E. Shnoll, I.A. Rubinshtein, K.I. Zenchenko, V.A. Shlekhtarev, A. V.Ka- Authors of articles published in Progress in Physics retain their rights to use their own minsky, A. A. Konradov, N. V.Udaltsova Experiments with Rotating articles in any other publications and in any Collimators Cutting out Pencil of α-Particles at Radioactive Decay of way they see fit. 239Pu Evidence Sharp Anisotropy of Space...... 81 F. Smarandache There Is No Speed Barrier for a Wave Phase Nor for Entang- This journal is powered by BaKoMa-TEX led Particles...... 85
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A New Method to Measure the Speed of Gravitation
Dmitri Rabounski E-mail: [email protected]
According to the standard viewpoint the speed of gravitation is the speed of weak waves of the metrics. This study proposes a new approach, defining the speed as the speed of travelling waves in the field of gravitational inertial force. D’Alembert’s equations of the field show that this speed is equal to the velocity of light corrected by gravitational potential. The approach leads to a new experiment to measure the speed of gravitation, which, using “detectors” such as planets and their satellites, is not linked to deviation of geodesic lines and quadrupole mass-detectors with their specific technical problems.
(0) 1 Introduction the Ricci tensor for the metric gαβ = gαβ + ζαβ is 1 ∂2ζ 1 Herein we use a pseudo-Riemannian space with the signature (0)μν αβ Rαβ = g μ ν = ζαβ , (+ ), where time is real and spatial coordinates are imag- 2 ∂x ∂x 2 −−− inary, because the projection of a four-dimensional impulse which simplifies Einstein’s field equations R 1 g R = αβ − 2 αβ on the spatial section of any given observer is positive = κ T + λg , where in this case R = g(0)μν R . In the − αβ αβ μν in this case. We also denote space-time indices in Greek, absence of matter and λ-fields (Tαβ = 0, λ = 0), that is, in while spatial indices are Roman. Hence the time term in (0) emptiness, the Einstein equations for the metric gαβ = gαβ + αβ d’Alembert’s operator = g α β will be positive, while + ζ become ∇ ∇ αβ β the spatial part (Laplace’s operator) will be negative Δ = ζα = 0 . = gik . − ∇i∇k Actually, these are the d’Alembert equations of the cor- By applying the d’Alembert operator to a tensor field, (0) rections ζαβ to the metric gαβ = gαβ + ζαβ (weak waves of we obtain the d’Alembert equations of the field. The non- the metric). Taking the flat wave travelling in the direction zero elements are the d’Alembert equations containing the x1 = x, we see field-inducing sources. The zero elements are the equations 2 2 1 ∂ ∂ β without the sources. If there are no sources the field is free, 2 2 2 ζα = 0 , 2 c ∂t − ∂x giving a free wave. There is the time term 1 ∂ containing a2 ∂t2 so weak waves of the metric travel at the velocity of light in the linear velocity a of the wave. For this reason, in the empty space. case of gravitational fields, the d’Alembert equations give This approach leads to an experiment, based on the prin- rise to a possibility of calculating the speed of propagation ciple that geodesic lines of two infinitesimally close test- of gravitational attraction (the speed of gravitation). At the particles will deviate in a field of waves of the metric. A same time the result may be different according to the way system of two real particles connected by a spring (a quadru- we define the speed as the velocity of waves of the metric, pole mass-detector) should react to the waves. Most of these or something else. experiments have since 1968 been linked to Weber’s detector. The usual approach sets forth the speed of gravitation as The experiments have not been technically decisive until follows [1, 5]. One considers the space-time metric gαβ = now, because of problems with precision of measurement (0) , composed of a Galilean metric (0) (wherein = gαβ + ζαβ gαβ and other technical problems [3] and some purely theoretical g(0) = 1, g(0) = 0, g(0) = δ ) and tiny corrections ζ de- 00 0i ik − ik αβ problems [4, 5]. fining a weak gravitational field. Because the ζαβ are tiny, we Is the approach given above the best? Really, the result- can raise and lower indices with the Galilean metric tensor ing d’Alembert equations are derived from that form of the (0) αβ gαβ. The quantities ζ are defined by the main property Ricci tensor obtained under the substantial simplifications of σβ β of the fundamental metric tensor gασg = δα as follows: higher order terms withheld (i .e. to first order). Eddington (0) σβ β αβ (gασ + ζασ) g = δα. Besides this approach defines g [1] wrote that a source of this approximation is a specific and g = det gαβ to within higher order terms withheld as reference frame which differs from Galilean reference frames αβ (0)αβk αβk (0) σ g = g ζ and g = g (1 + ζ), where ζ = ζσ . Be- by the tiny corrections ζαβ, the origin of which could be very − ...σ cause ζαβ are tiny we can take Ricci’s tensor Rαβ = Rασβ different from gravitation. This argument leads, as Eddington (the Riemann-Christoffel curvature tensor Rαβγδ contracted remarked, to a “vicious circle”. So the standard approach has on two indices) to within higher order terms withheld. Then inherent drawbacks, as follows:
D. Rabounski. A New Method to Measure the Speed of Gravitation 3 Volume 1 PROGRESS IN PHYSICS April, 2005
(1) The approach gives the Ricci tensor and hence the 1 ∗∂ + 2 vi are non-commutative. They are the chr.inv.-vector d’Alembert equations of the metric to within higher c ∂t of gravitational inertial force Fi, the chr.inv.-tensor of angular order terms withheld, so the velocity of waves of the velocities of the space rotation Aik, and the chr.inv.-tensor metric calculated from the equations is not an exact of rates of the space deformations Dik, namely theoretical result; (2) A source of this approximation are the tiny corrections 1 ∂w ∂vi Fi = , ζ to a Galilean metric, the origin of which may be √g ∂xi − ∂t αβ 00 very different: not only gravitation; (3) Two bodies attract one another because of the transfer 1 ∂vk ∂vi 1 Aik = + (Fivk Fkvi) , of gravitational force. A wave travelling in the field 2 ∂xi − ∂xk 2c2 − of gravitational force is not the same as a wave of the metric — these are different tensor fields. When a g0i w vi = c , √g00 = 1 2 , quadrupole mass-detector registers a signal, the detec- − √g00 − c tor reacts to a wave of the metric in accordance with ik √ this theory. Therefore it is concluded that quadrupole 1 ∗∂hik ik 1 ∗∂h k ∗∂ ln h Dik = ,D = ,D=Dk = , mass detectors would be the means by to discovery 2 ∂t −2 ∂t ∂t of waves of the metric. However, the experiment is where w is gravitational potential, vi is the linear velocity only incidental to the measurement of the speed of of the space rotation, h = g + 1 v v is the chr.inv.- gravitation. ik − ik c2 i k metric tensor, and also h = det h , g = √h g , For these reasons we are lead to consider gravitational k ikk − 00 g = det gαβ . Observable non-uniformity of the space is waves as waves travelling in the field of gravitational force, k k p i im p set up by the chr.inv.-Christoffel symbols Δjk = h Δjk,m, which provides two important advantages: α which are built just like Christoffel’s usual symbols Γμν = ασ (1) The mathematical apparatus of chronometric invariants = g Γμν,σ, using hik instead of gαβ. (physical observable quantities in the General Theory The four-dimensional generalization of the chr.inv.-quan- of Relativity) defines gravitational inertial force Fi tities Fi, Aik, and Dik had been obtained by Zelmanov [8] 2 β μ ν μ ν without the Riemann-Christoffel curvature tensor as F = 2c b a , A = chαh a , D = chαh d , α − βα αβ β μν αβ β μν [1, 2]. Using this method, we can deduce the exact where a = 1 ( b b ), d = 1 ( b + b ). αβ 2 ∇α β − ∇β α αβ 2 ∇α β ∇β α d’Alembert equations for the force field, giving an Following the study [9], we consider a field of the grav- exact formula for the velocity of waves of the force; 2 β itational inertial force Fα = 2c b aβα, the chr.inv.-spatial i − k (2) Experiments to register waves of the force field, using projection of which is F , so that Fi = hikF . The d’Alem- α 2 β α “detectors” such as planets or their satellites, does not bert equations of the vector field F = 2c b aβ∙ in the involve a quadrupole mass-detector and its specific absence of sources are − ∙ technical problems. F α = 0 . 2 The new approach Their chr.inv.-projections (referred to as the chr.inv.- d’Alembert equations) can be deduced as follows The basis here is the mathematical apparatus of chronometric invariants, created by Zelmanov in the 1940’s [1, 2]. Its b gαβ F σ = 0 , hi gαβ F σ = 0 . essence is that if an observer accompanies his reference body, σ ∇α∇β σ ∇α∇β his observable quantities (chronometric invariants) are pro- After some algebra we obtain the chr.inv.-d’Alembert jections of four-dimensional quantities on his time line and α equations for the field of the gravitational inertial force the spatial section, made by projecting operators bα = dx α 2 β α ds F = 2c b aβ∙ in their final form. They are and h = g +b b , which fully define his real reference − ∙ αβ − αβ α β space. Thus, chr.inv.-projections of a world-vector Qα are i m 1 ∗∂ k 1 ∗∂F k ∗∂F α Q0 i α i 2 FkF + 2 Fi + Dm k + bαQ = and hαQ = Q , while chr.inv.-projections of c ∂t c ∂t ∂x √g00 Q ∗∂ 2 a world-tensor of the 2nd rank Qαβ are bαbβQ = 00 , + hik [(D + A )F n] A F iF k + αβ g00 i kn kn 2 ik i ∂x − c iα β Q0 i k αβ ik h b Qαβ = , hαhβQ = Q . Physical observable 1 m m k n √g00 + 2 FmF D + ΔknDmF properties of the space are derived from the fact that the chr. c − ∗∂ 1 ∂ ∗∂ ∂ ik m n inv.-differential operators = and i = i + h Δik (Dmn + Amn) F = 0 , ∂t √g00 ∂t ∂x ∂x −
4 D. Rabounski. A New Method to Measure the Speed of Gravitation April, 2005 PROGRESS IN PHYSICS Volume 1
2 i 2 i k 1 ∗∂ F km ∗∂ F 1 i i ∗∂F producing ebbs and flows. An analogous “hump” follows h + Dk + Ak∙ + c2 ∂t2 − ∂xk∂xm c2 ∙ ∂t the Sun: its magnitude is more less. A satellite in an Earth i i orbit has the same ebb and flow oscillations — its orbit rises 1 ∗∂ i i k 1 ∗∂F 1 k ∗∂F + Dk + Ak∙ F + D + F + and falls a little, following the Moon and the Sun as well. c2 ∂t ∙ c2 ∂t c2 ∂xk A satellite in space experiences no friction, contrary of the 1 1 1 i i n k i i k m viscous waters of the oceans. A satellite is a perfect system, + 2 Dn +An∙ F D+ 4 FkF F + 2 ΔkmF F c ∙ c c − which reacts instantly to the flow. If the speed of gravitation ∂ hkm ∗ Δi F n + Δi Δn Δn Δi F p + is limited, the moment of the satellite’s maximum flow rise − ∂xk mn kn mp − km np should be later than the lunar/solar upper transit by the ∂F n ∂F i amount of time taken by waves of the gravitational force + Δi ∗ Δn ∗ = 0 . kn ∂xm − km ∂xn field to travel from the Moon/Sun to the satellite. The Earth’s gravitational field is not absolutely symmet- Calling upon the formulae for chr.inv.-derivatives, we ric, because of the imperfect form of the terrestrial globe. transform the first term in the chr.inv.-d’Alembert vector A real satellite reacts to the field defects during its orbital equations into the form flight around the Earth — the height of its orbit oscillates in 2 i 2 i i decimetres, giving rise to substantial noise in the experiment. 1 ∗∂ F 1 ∂ F 1 ∗∂w ∗∂F = + , For this reason a geostationary satellite would be best. Such c2 ∂t2 c2g ∂t2 c4√g ∂t ∂t 00 00 a satellite, having an equatorial orbit, requires an angular ve- so waves of gravitational inertial force travel at a velocity locity the same as that of the Earth. As a result, the height of k k 2 u , the square of which is uku = c g00 and the modulus a geostationary satellite above the Earth does not depend on non-uniformities of the Earth’s gravitational field. The height w k could be measured with high precision by a laser range- u = uku = c 1 2 . − c finder, almost without interruption, providing a possibility p Because waves of the field of gravitational inertial force of registering the moment of the maximum flow rise of the transfer gravitational interaction, this wave speed is the speed satellite, perfectly. of gravitation as well. The speed depends on the scalar In accordance with our formula the speed of gravita- potential w of the field itself, which leads us to the following tion near the Earth is 21 cm/sec less than the velocity of conclusions: light. In this case the maximum of the lunar flow wave in a (1) In a weak gravitational field, the potential w of which satellite orbit will be about 1 sec later than the lunar upper culmination. The lateness of the flow wave of the Sun will be is negligible but its gradient Fi is non-zero, the speed of gravitation equals the velocity of light; about 500 sec after the upper transit of the Sun. The question is how precisely could the moment of the maximum flow (2) According to this formula, the speed of gravitation will rise of a satellite in its orbit be determined, because the real be less than the velocity of light near bulky bodies maximum can be “fuzzy” in time. like stars or planets, where gravitational potential is perceptible. On the Earth’s surface slowing gravitation will be slower than light by 21 cm/sec. Gravitation 3 Effect of the curvature near the Sun will be about 6.3 104 cm/sec slower than × i light; If a space is homogeneous (Δkm = 0) and it is free of rotation 2 and deformation (Aik = 0, Dik = 0), then the chr.inv.- (3) Under gravitational collapse (w = c ) the speed of d’Alembert equations for the field of gravitational inertial gravitation becomes zero. force take the form Let us turn now from theory to experiment. An idea as i 1 ∗∂ k 1 ∗∂F to how to measure the speed of gravitation as the speed FkF + Fi = 0 , to transfer of the attracting force between space bodies had c2 ∂t c2 ∂t been proposed by the mathematician Dombrowski [10] in 1 ∂2F i ∂2F i 1 ∂F i 1 conversation with me more than a decade ago. But in the ∗ hkm ∗ + F k ∗ + F F kF i = 0 , c2 ∂t2 ∂xk∂xm c2 ∂xk c4 k absence of theory the idea had not developed to experiment − in that time. Now we have an exact formula for the speed so waves of gravitational inertial force are permitted even in of waves travelling in the field of gravitational inertial force, this very simple case. so we can propose an experiment to measure the speed (a Are waves of the metric possible in this case or not? Weber detector reacts to weak waves of the metric, so it does As it is known, waves of the metric are linked to the not apply to this experiment). space-time curvature derived from the Riemann-Christoffel The Moon attracts the Earth’s surface, causing the flow curvature tensor. If the first derivatives of the metric (the “hump” in the ocean surface that follows the moving Moon, space deformations) are zero, then its second derivatives
D. Rabounski. A New Method to Measure the Speed of Gravitation 5 Volume 1 PROGRESS IN PHYSICS April, 2005
(the curvature) are zero too. Therefore waves of the metric Acknowledgements have no place in a non-deforming space, while waves of gravitational inertial force are possible there. My late acknowledgements to Prof. Kyril Stanyukovich In connection with this fact, following the study [9], (1916–1989) and Dr. Abraham Zelmanov (1913–1987) for all another question arises. By how much does the curvature my experience in General Relativity given by them. Special affect waves of gravitational inertial force? thanks go to Dr. Kyril Dombrowski for our friendly convers- To answer the question let us recall that Zelmanov, follo- ations initiated this research many years ago. I am also very wing the same procedure by which the Riemann-Christoffel grateful to my closest colleagues: Dr. Larissa Borissova for tensor was introduced, after considering non-commutativity her checking of the calculations, and Stephen J. Crothers for of the chr.inv.-second derivatives of a vector ∗ ∗ Q his editing this paper. ∇i ∇k l − 2Aik ∗∂Ql ...j ∗ ∗ Q = + H Q , had obtained the chr. − ∇k ∇i l c2 ∂t lki j ...j References inv.-tensor Hlki like Schouten’s tensor [11]. Its generaliza- 1 tion gives the chr.inv.-curvature tensor Clkij = Hlkij 1. Eddington A. S. The mathematical theory of relativity. 4 − H + H H , which has all the properties of the Cambridge University Press, Cambridge, 1924 (ref. with the jkil klji iljk −Riemann-Christoffel− tensor in the observer’s spatial section. 3rd exp. edition, GTTI, Moscow, 1934). So the chr.inv.-spatial projection Ziklj = c2Riklj of the 2. Landau L. D. and Lifshitz E. M. The classical theory of fields. − Riemann-Christoffel tensor Rαβγδ, after contraction twice by GITTL, Moscow, 1939 (ref. with the 4th final exp. edition, il ik 2 ik 2 hik, is Z = h Zil = DikD D AikA c C, where Butterworth–Heinemann, 1980). j lj −...i − im − C = Cj = h Clj and Ckj = Ckij = h Ckimj [1]. 3. Rudenko V.N. Relativistic experiments in gravitational field. At the same time, as Synge’s∙ well-known book [12] Uspekhi Fizicheskikh Nauk, 1978, v. 126 (3), 361–401. shows, in a space of constant four-dimensional curvature, 4. Borissova L. B. Relative oscillations of test-particles in K = const, we have Rαβγδ = K (gαγ gβδ gαδ gβγ ), Rαβ = accompanying reference frames. Doklady Acad. Nauk USSR, = 3Kg , R = 12K. With these formulae− as a basis, after 1975, v. 225 (4), 786–789. − αβ − calculation of the chr.inv.-spatial projection of the Riemann- 5. Borissova L. B. Quadrupole mass-detector in a field of weak Christoffel tensor, we deduce that in a constant curvature flat gravitational waves. Izvestia VUZov, Physica, 1978 (10), space Z = 6c2K. Equating this to the same quantity in an 109–114. arbitrary curvature space, we obtain a correlation between 6. Zelmanov A. L. Chronometric invariants. Dissertation, 1944. the four-dimensional curvature K and the observable three- First published: CERN, EXT-2004-117, 236 pages. dimensional curvature in the constant curvature space 7. Zelmanov A. L. Chronometric invariants and co-moving coordinates in the general relativity theory. Doklady Acad. 6c2K = D Dik D2 A Aik c2C. ik − − ik − Nauk USSR, 1956, v. 107 (6), 815–818. If the four-dimensional curvature is zero (K = 0), and 8. Zelmanov A. L. Orthometric form of monad formalism and its relations to chronometric and kinemetric invariants. Doklady the space does no deformations (D = 0 — its metric is ik Acad. Nauk USSR, 1976, v. 227 (1), 78–81. stationary, hik = const), then no waves of the metric are possible. In such a space the observable three-dimensional 9. Rabounski D. D. The new aspects of General Relativity. CERN, curvature is EXT-2004-025, 117 pages. 1 ik 10. Dombrowski K. I. Private communications, 1990. C = AikA , −c2 11. Schouten J. A. und Struik D. J. Einfuhrung¨ in die neuren Meth- oden der Differentialgeometrie. Zentralblatt fur¨ Mathematik, which is non-zero (C = 0), only if the space rotates (Aik = 0). If aside of these factors,6 the space does not rotate, then6 its 1935, Bd. 11 und Bd. 19. observable curvature also becomes zero; C = 0. Even in this 12. Synge J. L. Relativity: the General Theory. North Holland, case the chr.inv.- d’Alembert equations show the presence of Amsterdam, 1960 (referred with the 2nd edition, Foreign waves of gravitational inertial force. Literature, Moscow, 1963, 432 pages). What does this imply? As a matter of fact, gravitational attraction is an everyday reality in our world, so waves of gravitational inertial force transferring the attraction shall be incontrovertible. Therefore we adduce the alternatives: (1) Waves of gravitational inertial force depend on a cur- vature of space — then the real space-time is not a space of constant curvature, or, (2) Waves of gravitational inertial force do not depend on the curvature.
6 D. Rabounski. A New Method to Measure the Speed of Gravitation April, 2005 PROGRESS IN PHYSICS Volume 1
Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
There can be generated many paradoxes or quasi-paradoxes that may occur from the combination of quantum and non-quantum worlds in physics. Even the passage from the micro-cosmos to the macro-cosmos, and reciprocally, can generate unsolved questions or counter-intuitive ideas. We define a quasi-paradox as a statement which has a prima facie self-contradictory support or an explicit contradiction, but which is not completely proven as a paradox. We present herein four elementary quantum quasi-paradoxes and their corresponding quantum Sorites paradoxes, which form a class of quantum quasi-paradoxes.
1 Introduction stable/unstable, long time living/short time living, as well as on the fact that there is not a clear separation between these According to the Dictionary of Mathematics (Borowski and pairs of antinomies. Borwein, 1991 [1]), the paradox is “an apparently absurd or self-contradictory statement for which there is prima facie 2.1.1 Invisible Quasi-Paradox: Our visible world is com- support, or an explicit contradiction derived from apparently posed of a totality of invisible particles. unexceptionable premises”. Some paradoxes require the revi- 2.1.2 Invisible Sorites Paradox: There is not a clear frontier sion of their intuitive conception (Russell’s paradox, Cantor’s between visible matter and invisible matter. paradox), others depend on the inadmissibility of their de- (a) An invisible particle does not form a visible ob- scription (Grelling’s paradox), others show counter-intuitive ject, nor do two invisible particles, three invisible features of formal theories (Material implication paradox, particles, etc. However, at some point, the collec- Skolem Paradox), others are self-contradictory — Smarand- tion of invisible particles becomes large enough ache Paradox: “All is the
F. Smarandache. Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes 7 Volume 1 PROGRESS IN PHYSICS April, 2005
living” matter is formed by very “short-time-living” elementary particles. 2.4.2 Short-Time-Living Sorites Paradox: Similarly, there is not a clear frontier between the “long-time-living” matter and the “short-time-living” matter.
3 Conclusion
“More such quantum quasi-paradoxes and paradoxes can be designed, all of them forming a class of Smarandache quantum quasi-paradoxes.” (Dr. M. Khoshnevisan, Griffith University, Gold Coast, Queensland, Australia [3])
References
1. Borowski E. J. and Borwein J. M. The Harper Collins Diction- ary of Mathematics. Harper Perennial, A Division of Harper Collins Publishers, New York, 1991. 2. Weisstein E. W. Smarandache Paradox. CRC Concise Encyclo- paedia of Mathematics, CRC Press, Boca Raton, Florida, 1998, 1661, (see the e-print version of this article in http://mathworld. wolfram.com/SmarandacheParadox.html). 3. Khoshnevisan M. Private communications, 1997. 4. Motta L. A Look at the Smarandache Sorites Paradox. Proceed- ings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi-bin/amca/caft-04). 5. Niculescu G. On Quantum Smarandache Paradoxes. Proceed- ings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi-bin/amca/caft-20). 6. Smarandache F. Invisible paradox. Neutrosophy, Neutrosophic Probability, Set, and Logic, American Research Press, Reho- both, 1998 (see the third e-print edition of the book in the web site http://www.gallup.unm.edu/ smarandache). ∼ 7. Smarandache F. Sorites paradoxes. Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry (edited by M. L. Perez), Xiquan Pub- lishing House, Phoenix, 2000.
8 F. Smarandache. Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes April, 2005 PROGRESS IN PHYSICS Volume 1
A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
Besides matter and antimatter there must exist unmatter (as a new form of matter) in accordance with the neutrosophy theory that between an entity and its opposite
1 Introduction Etymologically “un-matter” comes from [ME
F. Smarandache. A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles 9 Volume 1 PROGRESS IN PHYSICS April, 2005
artificially made tritium (3H) which has two neutrons have We can collide unmatter 1 with unmatter 2, or unmatter corresponding unmatter atoms of both types, un-deuterium 1 with antimatter, or unmatter 2 with matter. and un-tritium respectively. The isotopes of an element X When two, three, or four of them (unmatter 1, unmatter 2, differ in the number of neutrons, thus their nuclear mass is matter, antimatter) collide together, they annihilate and turn different, but their nuclear charges are the same. into energy which can materialize at high energy into new For all other matter atom X, there is corresponding an particles and antiparticles. antimatter atom and two unmatter atoms The unmatter atoms are also neutral for the same reason 4 Existence of unmatter that either the number of electrons is equal to the number of protons in the first type, or the number of antielectrons is The existence of unmatter in the universe has a similar chance equal to the number of antiprotons in the second type. to that of the antimatter, and its production also difficult for If antimatter exists then a higher probability would be for present technologies. At CERN it will be possible to test the the unmatter to exist, and reciprocally. production of unmatter. Unmatter atoms of the same type stick together form If antimatter exists then a higher probability would be for an unmatter molecule (we call it unmolecule), and so on. the unmatter to exist, and reciprocally. Similarly one has two types of unmatter molecules. The 1998 Alpha Magnetic Spectrometer (AMS) flown on The isotopes of an atom or element X have the same the International Space Station orbiting the Earth would be atomic number (same number of protons in the nucleus) able to detect, besides cosmic antimatter, unmatter if any. but different atomic masses because the different number of neutrons. Therefore, similarly the un-isotopes of type 1 of X will 5 Experiments be formed by electrons, protons, and antineutrons, while the un-isotopes of type 2 of X will be formed by antielectrons, Besides colliding electrons, or protons, would be interesting antiprotons, and neutrons. in colliding neutrons. Also, colliding a neutron with an anti- An ion is an atom (or group of atoms) X which has neutron in accelerators. last one or more electrons (and as a consequence carries a We think it might be easier to produce in an experiment negative charge, called anion, or has gained one or more an unmatter atom of deuterium (we can call it un-deuterium electrons (and as a consequence carries a positive charge, of type 1). The deuterium, which is an isotope of the ordinary called cation). hydrogen, has an electron, a proton, and a neutron. The Similarly to isotopes, the un-ion of type 1 (also called idea would be to convert/transform in a deuterium atom the un-anion 1 or un-cation 1 if resulted from a negatively neutron into an antineutron, then study the properties of the or respectively positive charge ion) of X will be formed resulting un-deuterium 1. by electrons, protons, and antineutrons, while the un-ion of Or, similarly for un-deuterium 2, to convert/transform in type 2 of X (also called un-anion 2 or un-cation 2 if resulted a deuterium atom the electron into an antielectron, and the from a negatively or respectively positive charge ion) will be proton into an antiproton (we can call it un-deuterium of formed by antielectrons, antiprotons, and neutrons. type 2). The ion and the un-ion of type 1 have the same charges, Or maybe choose another chemical element for which while the ion and un-ion of type 2 have opposite charges. any of the previous conversions/transformations might be possible. D. Nonmatter means what is not matter, therefore non- matter actually comprises antimatter and unmatter. Similarly one defines a nonnucleus. 6 Neutrons and antineutrons Hadrons consist of baryons and mesons and interact via 3 Unmatter propulsion strong force. Protons, neutrons, and many other hadrons are composed We think (as a prediction or supposition) it could be possible from quarks, which are a class of fermions that possess at using unmatter as fuel for space rockets or for weapons a fractional electric charge. For each type of quark there platforms because, in a similar way as antimatter is presup- exists a corresponding antiquark. Quarks are characterized posed to do [4, 5], its mass converted into energy will be by properties such as flavor (up, down, charm, strange, top, fuel for propulsion. or bottom) and color (red, blue, or green). It seems to be a little easier to build unmatter than A neutron is made up of quarks, while an antineutron is antimatter because we need say antielectrons and antiprotons made up of antiquarks. only (no need for antineutrons), but the resulting energy A neutron (see [9]) has one Up quark (with the charge 2 19 might be less than in matter-antimatter collision. of + 3 ×1.606×10 C) and two Down quarks (each with the
10 F. Smarandache. A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles April, 2005 PROGRESS IN PHYSICS Volume 1
1 19 charge of 3 ×1.606×10 C), while an antineutron has one 3. Webster’s New World Dictionary. Third College Edition, − 2 19 Simon and Schuster Inc., 1988. anti Up quark (with the charge of 3 ×1.606×10 C) and − 1 4. Mondardini R. The history of antimatter. CERN Laboratory, two anti Down quarks (each with the charge of + 3 ×1.606× 19 Geneve,` on-line http://livefromcern.web.cern.ch/livefromcern/ ×10 C). An antineutron has also a neutral charge, through it is antimatter/history/AM-history00.html. opposite to a neutron, and they annihilate each other when 5. De Rujula´ A. and Landua R. Antimatter — frequently asked meeting. questions. CERN Laboratory, Geneve,` http://livefromcern.web. Both, the neutron and the antineutron, are neither attract- cern.ch/livefromcern/antimatter/FAQ1.html. ed to nor repelling from charges particles. 6. Pompos A. Inquiring minds — questions about physics. Fermi- lab, see on-line http://www.fnal.gov/pub/inquiring/questions/ antineuron.html. 7 Characteristics of unmatter
Unmatter should look identical to antimatter and matter, also the gravitation should similarly act on all three of them. Unmatter may have, analogously to antimatter, utility in medicine and may be stored in vacuum in traps which have the required configuration of electric and magnetic fields for several months.
8 Open Questions
8.a Can a matter atom and an unmatter atom of first type stick together to form a molecule? 8.b Can an antimatter atom and an unmatter atom of sec- ond type stick together to form a molecule? 8.c There might be not only a You and an anti-You, but some versions of an un-You in between You and anti- You. There might exist un-planets, un-stars, un- galaxies? There might be, besides our universe, an anti-universe, and more un-universes? 8.d Could this unmatter explain why we see such an im- balance between matter and antimatter in our corner of the universe? (Jeff Farinacci) 8.e If matter is thought to create gravity, is there any way that antimatter or unmatter can create antigravity or ungravity? (Mike Shafer from Cornell University) I assume that since the magnetic field or the gravitons generate gravitation for the matter, then for antimatter and unmatter the corresponding magnetic fields or gravitons would look different since the charges of subatomic particles are different. . . I wonder how would the universal law of attraction be for antimmater and unmatter?
References
1. Smarandache F. Unmatter, mss., 1980, Archives Valcea.ˆ 2. Smarandache F. A unifying field in logics: neutrosophic logic. Neutrosophy, Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth, 2002, 144 p. (see it in e-print: http://www.gallup.unm.edu/ smarandache). ∼
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Fractality Field in the Theory of Scale Relativity
Laurent Nottale Centre National de la Recherche Scientifique, Laboratoire “Univers et Theories”´ , Observatoire de Paris-Meudon, F-92195 Meudon Cedex, France E-mail: [email protected]
In the theory of scale relativity, space-time is considered to be a continuum that is not only curved, but also non-differentiable, and, as a consequence, fractal. The equation of geodesics in such a space-time can be integrated in terms of quantum mechanical equations. We show in this paper that the quantum potential is a manifestation of such a fractality of space-time (in analogy with Newton’s potential being a manifestation of curvature in the framework of general relativity).
1 Introduction Let us explicitly introduce the real and imaginary parts of the complex velocity = V iU, The theory of scale relativity aims at describing a non- V − differentiable continuous manifold by the building of new d ∂ V = +V i U + Δ (V iU) = 0. tools that implement Einstein’s general relativity concepts dt ∂t ∙∇ − { ∙∇ D } − in the new context (in particular, covariant derivative and (3) geodesics equations). We refer the reader to Refs. [1, 2, 3, 4] We see in this expression that the real part of the covar- for a detailed description of the construction of these tools. iant derivative, dR/dt = ∂/∂t + V , is the standard total ∙∇ In the present short research note, we want to address a derivative expressed in terms of partial derivatives, while specific point of the theory, namely, the emergence ofan the new terms are included in the imaginary part, dI /dt = additional potential energy which manifests the fractal and = (U + Δ). The field will find its origin in the conse- − ∙∇ D nondifferentiable geometry. quences of these additional terms on the imaginary part of the velocity U. Indeed, by separating the real and imaginary parts, equation− (3) reads: 2 Non relativistic quantum mechanics ∂ 2.1 Quantum potential + V V (U + Δ)U ∂t ∙∇ − ∙∇ D − In the scale relativity approach, one decomposes the velocity (4) ∂ field on the geodesics bundle of a nondifferentiable space- i (U + Δ)V + + V U = 0 . − ∙∇ D ∂t ∙∇ time in terms of a classical, differentiable part, , and of a fractal, divergent, nondifferentiable part of zeroV mean. W Therefore the real part of this equation takes the form of Both velocity fields are complex due to a fundamental two- an Euler-Newton equation of dynamics valuedness of the classical (differentiable) velocity issued from the nondifferentiability [1]. Then one builds a complex ∂ + V V = (U + Δ)U, (5) covariant total derivative that reads in the simplest case ∂t ∙∇ ∙∇ D (spinless particle, nonrelativistic velocities and no external i. e., field) [1, 2, 3] dV F d ∂ = , (6) = + i Δ . (1) dt m dt ∂t V∙∇ − D where the total derivative of the velocity field V takes its The constant 2 =
12 L. Nottale. Fractality Field in the Theory of Scale Relativity April, 2005 PROGRESS IN PHYSICS Volume 1
Let us now show that the additional force derives from a bodies should be injected in the energy-momentum tensor, potential. Indeed, the imaginary part of the complex velocity so that this is a looped system which has no exact analytical field is given, in terms of the modulus of ψ, by the expression: solution. In the case of a quantum mechanical particle considered U = ln P. (8) in scale relativity, the loop between the motion (geodesics) D∇ equation and the field equation is even more tight. Indeed, The force becomes here the concept of test-particle loses its meaning. Even in the case of only one “particle”, the space-time geometry is F = m 2 [( ln P )( ln P ) + Δ( ln P )] . (9) D ∇ ∙∇ ∇ ∇ determined by the particle itself and by its motion, so that the Now, by introducing √P in this expression, one makes field equation and the geodesics equation now participate of explicitly appear the remarkable identity that is already at the the same level of description. This explains why the motion/ heart of the proof of the Schrodinger¨ equation ([1], p. 151), geodesics equation, in its Hamilton-Jacobi form that takes the namely, form of the Schrodinger¨ equation, is obtained without having first written the field equation in an explicit way. Actually, F = 2m 2 2( ln √P )( ln √P ) + the potential Q is implicitly contained in the Schrodinger¨ D ∇ ∙∇ ∇ form of the equations, and it is made explicit only when h (10) Δ√P coming back to a fluid-like Euler-Newton representation. + Δ( ln √P ) = 2m 2 . In the end, the particle is described by a wave function ∇ D ∇ √P ! i (which is constructed, in the scale relativity theory, from the geodesics), of which only the square of the modulus Therefore the force F derives from a potential energy P is observable. Therefore one expects the “field” to be given Δ√P by a function of P , which is exactly what is found. Q = 2m 2 , (11) − D √P 2.2 Invariants and energy balance which is nothing but the standard “quantum potential”, but here established as a mere manifestation of the nondifferen- Let us now make explicit the energy balance by accounting tiable and fractal geometry instead of being deduced from a for this additional potential energy. This question has already postulated Schrodinger¨ equation. been discussed in [7, 8] and in [9], but we propose here a The real part of the motion equation finally takes the different presentation. We shall express the energy equation standard form of the equation of dynamics in presence of a in terms of the various equivalent variables which we use in scalar potential, scale relativity, namely, the wave function ψ, the complex velocity or its real and imaginary parts V and U. V − dV ∂ Q The first and main form of the energy equation isthe = + V V = ∇ , (12) dt ∂t ∙∇ − m Schrodinger¨ equation itself, that we have derived as a prime integral of the geodesics equation. The Schrodinger¨ equation while the imaginary part is the equation of continuity ∂P/∂t+ is therefore the quantum equivalent of the metric form (i. e., + div(PV ) = 0. The fact that the field equation is derived of the equation of conservation of the energy). It may be from the same remarkable identity that gives rise to the written in the free case under the form Schrodinger¨ equation is also manifest in the similarity of its Δψ ∂ ln ψ form with the free stationary Schrodinger¨ equation, namely, 2 = i . (14) D ψ − D ∂t Q E 2Δ√P + √P = 0 2Δψ + ψ = 0 . (13) In the stationary case with given energy E, it becomes: D 2m ←→ D 2m Δψ E = 2m 2 . (15) Now, the form (11) of the field equation means that the − D ψ field can be known only after having solved the Schrodinger¨ equation for the wave function. This is a situation somewhat Now we can use the fundamental remarkable identity different from that of general relativity, where, at least for Δψ/ψ = ( ln ψ)2 + Δ ln ψ. Re-introducing the complex ∇ test-particles, the description is reversed: given the energy- velocity field = 2i ln ψ in this expression we finally V − D∇ momentum tensor, one solves the Einstein field (i. e. space- obtain the correspondence: time geometry) equations for the metric potentials, then one Δψ 1 writes the geodesics equation in the space-time so determined E = 2m 2 = m 2 2i . (16) − D ψ 2 V − D∇∙V and solve it for the motion of the particle. However, even in general relativity this case is an ideally simplified situation, Note that when a potential term is present, all these since already in the two-body problem the motion of the relations remain true by replacing E by E φ. −
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This is the non-relativistic equivalent of Pissondes’ rela- particular, in the standard QM case, λ is the Compton length μ μ tion [8] in the relativistic case, μ + iλ∂ μ = 1 (see also of the particle, λ = ~/mc, and we recover = ~/2m. hereafter). Therefore the form ofV theV energy VE = (1/2)mV 2 The calculations are similar to the non-relativisticD case. is not conserved: this is precisely due to the existence of the We decompose the complex velocity in terms of its real additional potential energy of geometric origin. Let us prove and imaginary parts, α = Vα i Uα, so that the geodesics this statement. equation becomes V − From equation (16) we know that the imaginary part of 2 λ ( 2i ) is zero. By writing its real part in terms of V μ i U μ ∂μ ∂ (V i U ) = 0 , (24) V − D∇∙V − − 2 μ α − α the real velocities U and V , we find: 1 i. e., E = m 2 2i = 2 V − D∇∙V (17) μ μ λ μ 1 2 2 V ∂ V U ∂ ∂ U = m (V U 2 U) . μ α − − 2 μ α − 2 − − D∇∙ (25) λ Now we can express the potential energy Q given in i U μ ∂μ ∂ V + V μ∂ U = 0 . − − 2 μ α μ α equation (11) in terms of the velocity field U: 1 The real part of this equation takes the form of a relativ- Q = m (U 2 2 U) , (18) −2 − D∇∙ istic Euler-Newton equation of dynamics: so that we finally write the energy balance under the three dV λ α = V μ∂ V = U μ ∂μ ∂ U . (26) equivalent forms: ds μ α − 2 μ α Δψ E = 2m 2 = Therefore the relativistic case is similar to the non-relativ- − D ψ (19) istic one, since a generalized force also appears in the right- 1 1 hand side of this equation. Let us now prove that it also = m 2 2i = mV 2 + Q. 2 V − D∇∙V 2 derives from a potential. Using the expression for Uα in terms of the modulus √ of the wave function, More generally, in presence of an external potential en- P ergy φ and in the non-stationary case, it reads: U = λ ∂ ln √P, (27) α − α ∂S 1 R = mV 2 + Q + φ , (20) we may write the force under the form − ∂t 2 Fα μ where SR is the real part of the complex action (i. e., = λ ∂ ln √P ∂ ( λ ∂ ln √P ) + m − μ − α SR/2m is the phase of the wave function). D 2 λ μ + ∂ ∂μ∂α ln √P = (28) 3 Relativistic quantum mechanics 2 1 = λ2 ∂μ ln √P ∂ ∂ ln √P + ∂μ∂ ∂ ln √P . 3.1 Quantum potential μ α 2 μ α All the above description can be directly generalized to μ μ Since ∂ ∂μ∂α = ∂α∂ ∂μ commutes and since relativistic QM and the Klein-Gordon equation [10, 2, 3]. μ μ μ ∂α(∂ ln f ∂μ ln f) = 2 ∂ ln f ∂α∂ ln f, we obtain The geodesics equation still reads in this case: F 1 d α = λ2 ∂ ∂μ ln √P ∂ ln √P +∂μ∂ ln √P . (29) Vα = 0 , (21) m 2 α μ μ ds We can now make use of the remarkable identity (that where the total derivative is given by [10, 3] generalizes to four dimensions the one which is also at the heart of the non-relativistic case) d μ λ μ = + i ∂ ∂μ . (22) ds V 2 μ μ μ ∂ ∂μ√P ∂ ln √P ∂μ ln √P + ∂ ∂μ ln √P = , (30) The complex velocity field α reads in terms of the wave √P function V and we finally obtain = iλ ∂ ln ψ . (23) Vα α μ The relation between the non-relativistic fractal param- dVα 1 2 ∂ ∂μ√P = λ ∂α . (31) eter and the relativistic one λ is simply 2 = λc. In ds 2 √P D D !
14 L. Nottale. Fractality Field in the Theory of Scale Relativity April, 2005 PROGRESS IN PHYSICS Volume 1
Therefore, as in the non-relativistic case, the force derives One obtains: from a potential energy μ μ μ V Vμ (U Uμ λ ∂ Uμ) = 1 , μ − − 1 2 2 ∂ ∂μ√P (39) QR = mc λ , (32) 2 V μU λ ∂μV = 0 . 2 √P μ − μ that can also be expressed in terms of the velocity field U as Then the energy balance writes, in terms of the additional potential energy QR 1 2 μ μ QR = mc (U Uμ λ ∂ Uμ) . (33) 2 − QR μ (40) V Vμ = 1 + 2 2 . At the non-relativistic limit (c ), the Dalembertian mc → ∞ ∂μ∂ = (∂2/c2∂t2 Δ) is reduced to Δ, and since λ = Let us show that we actually expect such a relation for the μ − − = 2 /c, we recover the nonrelativistic potential energy Q = quadratic invariant in presence of an external potential φ. The D = 2m 2Δ√P/√P . Note the correction to the potential energy relation writes in this case (E φ)2 = p2c2 + m2c4, − D introduced by Pissondes [7] which is twice this potential and i. e. E2 p2c2 = m2c4 + 2E φ φ2.− Introducing the rest − −2 therefore cannot agree with the nonrelativistic limit. frame energy by writing E = mc + E0, we obtain
E2 p2c2 3.2 Invariants and energy balance V μV = − = μ m2c4 (41) 2 As shown by Pissondes [7, 8], the four-dimensional energy φ E0 φ φ μ = 1 + 2 + 2 . equation u uμ = 1 is generalized in terms of the complex mc2 mc2 mc2 m2c4 μ μ − velocity under the form μ + iλ∂ μ = 1. Let us show that the additional term isV aV manifestationV of the new scalar This justifies the relativistic factor 2 in equation (40)and field Q which takes its origin in the fractal and nondifferen- supports the interpretation of QR in terms of a potential, at tiable geometry. Start with the geodesics equation least at the level of the leading terms. Now, concerning the additional terms, it should remain d λ Vα = μ + i ∂μ ∂ = 0 . (34) clear that this is only an approximate description in terms of ds V 2 μ Vα field theory of what are ultimately (in this framework) the manifestations of the fractal and nondifferentiable geometry Then, after introducing the wave function by using the of space-time. Therefore we expect the field theory descript- relation = iλ ∂ ln ψ, after calculations similar to the α α ion to be a first order approximation in the same manner as, above onesV (now on the full function ψ instead of only its in general relativity, the description in terms of Newtonian modulus √P ), the geodesics equation becomes: potential. 2 In particular, in the non-relativistic limit c the last d α λ μ μ → ∞ V = ∂α (∂ ln ψ ∂μ ln ψ + ∂ ∂μ ln ψ) = two terms of equation (41) vanish and we recover the energy ds − 2 (35) equation (19) which is therefore exact in this case. 1 ∂μ∂ ψ = ∂ λ2 μ = 0 . 2 α − ψ 4 Conclusion Under its right-hand form, this equation is integrated in terms of the Klein-Gordon equation, Placing ourselves in the framework of the scale-relativity theory, we have shown in a detailed way that the quantum 2 μ λ ∂ ∂μψ + ψ = 0 . (36) potential, whose origin remained mysterious in standard quantum mechanics, is a manifestation of the nondifferen- Under its left hand form, the integral writes tiability and fractality of space-time in the new approach. This result is expected to have many applications, as λ2(∂μ ln ψ ∂ ln ψ + ∂μ∂ ln ψ) = 1 . (37) − μ μ well in physics as in other sciences, including biology [4]. It becomes in terms of the complex velocity [8] It has been used, in particular, to suggest a new solution to the problem of “dark matter” in cosmology [11, 5], based μ μ μ + iλ∂ μ = 1 , (38) on the proposal that chaotic gravitational system can be V V V described on long time scales (longer than their horizon of which is therefore but another form taken by the KG equation predictibility) by the scale-relativistic equations and therefore (as expected from the fact that the KG equation is the by a macroscopic Schrodinger¨ equation [12]. In this case quantum equivalent of the Hamilton-Jacobi equation). Let there would be no need for additional non baryonic dark us now separate the real and imaginary parts of this equation. matter, since the various observed non-Newtonian dynamical
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effects (that the hypothesis of dark matter wants to explain despite the check of all attempts of detection) would be readily accounted for by the new scalar field that manifests the fractality of space.
References
1. Nottale L. Fractal space-time and microphysics: towards a Theory of Scale Relativity. World Scientific, Singapore, 1993. 2. Nottale L. Chaos, Solitons & Fractals, v. 7, 1996, 877. 3. Cel´ erier´ M. N. and Nottale L. J. Phys. A: Math. Gen., v. 37, 2004, 931. 4. Nottale L. Computing Anticipatory Systems. CASYS’03 — Sixth International Conference, Liege,` Belgique, 11–16 August 2003, Ed. by Daniel M. Dubois, American Institute of Physics Conference Proceedings, v. 718, 2004, p. 68. 5. da Rocha D. and Nottale L. Chaos Solitons & Fractals, v. 16, 2003, 565. 6. Nottale L. Chaos, Solitons & Fractals, v. 10, 1999, 459. 7. Pissondes J. C. Chaos, Solitons & Fractals, v. 10, 1999, 513. 8. Pissondes J. C. J. Phys. A: Math. Gen., v. 32, 1999, 2871. 9. Nottale L. Chaos, Solitons & Fractals, v. 9, 1998, 1051. 10. Nottale L. Relativity in General, 1993 Spanish Relativity Meeting, Salas, Ed. by J. Diaz Alonso and M. Lorente Paramo (Frontieres),` 1994, p. 121. 11. Nottale L. Frontiers of Fundamental Physics, Proceedings of Birla Science Center Fourth International Symposium, 11– 13 Dec. 2000, Hyderabad, India, Ed. by B. G. Sidharth and M. V. Altaisky, Kluwer Academic, p. 65. 12. Nottale L. Astron. Astrophys., v. 327, 1997, 867.
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On the Possibility of Instant Displacements in the Space-Time of General Relativity
Larissa Borissova and Dmitri Rabounski E-mail: [email protected]; [email protected]
Employing the mathematical apparatus of chronometric invariants (physical observable quantities), this study finds a theoretical possibility for the instant displacement of particles in the space-time of the General Theory of Relativity. This is to date the sole theoretical explanation of the well-known phenomenon of photon teleportation, given by the purely geometrical methods of Einstein’s theory.
As it is known, the basic space-time of the General The- Massless particles are related to light-like particles — quanta ory of Relativity is a four-dimensional pseudo-Riemannian of electromagnetic fields (photons). space, which is, in general, inhomogeneous, curved, rotating, A condition under which a particle may realize an instant and deformed. There the square of the space-time interval displacement (teleportation) is equality to zero of the observ- 2 α β ds = gαβ dx dx , being expressed in the terms of physical able time interval dτ = 0 so that the teleportation condition is observable quantities — chronometric invariants [1, 2], takes i 2 the form w + viu = c , 2 2 2 2 ds = c dτ dσ . i − where ui = dx is its three-dimensional coordinate velocity. Here the quantity dt From this the square of that space-time interval by which w 1 this particle is instantly displaced takes the form dτ = 1 dt v dxi c2 c2 i − − 2 2 2 w 2 2 i k 2 ds = dσ = 1 c dt + g dx dx , is an interval of physical observable time, w=c (1 √g00) is − − − c2 ik g − the gravitational potential, v = c 0i is the linear velocity i i √g w viu − 00 where 1 2 = 2 in this case, because dτ = 0. 2 i k − c c of the space rotation, dσ = hik dx dx is the square of Actually, the signature (+ ) in the space-time area 1 −−− a spatial observable interval, hik = gik + vivk is the of a regular observer becomes ( +++) in that space-time − c2 − metric observable tensor, gik are spatial components of the area where particles may be teleported. So the terms “time” fundamental metric tensor gαβ (space-time indices are Greek and “three-dimensional space” are interchanged in that area. α, β = 0, 1, 2, 3, while spatial indices — Roman i, k =1, 2, 3). “Time” of teleporting particles is “space” of the regular Following this form we consider a particle displaced by observer, and vice versa “space” of teleporting particles is ds in the space-time. We write ds2 as follows “time” of the regular observer. Let us first consider substantial particles. As it easy to v2 ds2 = c2dτ 2 1 , see, instant displacements (teleportation) of such particles − c2 manifests along world-trajectories in which ds2= dσ2=0 − 6 i is true. So the trajectories represented in the terms of observ- where v2 = h vivk, and vi =dx is the three-dimensional ik dτ able quantities are purely spatial lines of imaginary three- observable velocity of the particle. So ds is: (1) a substantial dimensional lengths dσ, although being taken in ideal world- quantity under v < c; (2) a zero quantity under v = c; (3) an coordinates t and xi the trajectories are four-dimensional. In imaginary quantity under v > c. a particular case, where the space is free of rotation (vi = 0) Particles of non-zero rest-masses m = 0 (substance) can 0 6 or its rotation velocity vi is orthogonal to the particle’s co- be moved: (1) along real world-trajectories cdτ > dσ, having i i i i ordinate velocity u (so that viu = vi u cos (vi; u ) = 0), real relativistic masses m= m0 ; (2) along imagin- | || | 2 2 substantial particles may be teleported only if gravitational 1 v /c 2 ary world-trajectories cdτ < dσ, having− imaginary relativistic collapse occurs (w = c ). In this case world-trajectories of teleportation taken in ideal world-coordinates become also im0 p masses m = (tachyons). World-lines of both 2 i k v2/c2 1 purely spatial ds = gik dx dx . kinds are known as non-isotropic− trajectories. Second, massless light-like particles (photons) may be p Particles of zero rest-masses m0 = 0 (massless particles), teleported along world-trajectories located in a space of the having non-zero relativistic masses m=0, move along world- metric 6 trajectories of zero four-dimensional lengths cdτ = dσ at the w 2 ds2 = dσ2 = 1 c2dt2 + g dxidxk = 0 , velocity of light. They are known as isotropic trajectories. − − − c2 ik
L. Borissova and D. Rabounski. On the Possibility of Instant Displacements in the Space-Time of General Relativity 17 Volume 1 PROGRESS IN PHYSICS April, 2005
because for photons ds2 = 0 by definition. So the space of formulate regular derivatives through chronometrically in- photon teleportation characterizes itself by the conditions variant (physical observable) derivatives ∗∂ = 1 ∂ and ds2 = 0 and dσ2 = c2dτ 2 = 0. ∂t √g00 ∂t The obtained equation is like the “light cone” equation ∗∂ = ∂ + 1 v ∗∂ and we use g00 = 1 1 1 v vi , i i 2 i g00 2 i 2 2 2 ∂x ∂x c ∂t − c c dτ dσ = 0 (dσ = 0, dτ = 0), elements of which are i i 0i ik ik − 6 6 vk = hikv , v = cg √g00, g = h . world-trajectories of light-like particles. But, in contrast to The eikonal equation− in zero-space− takes the form the light cone equation, the obtained equation is built by ideal world-coordinates t and xi — not this equation in the ∗∂ψ ∗∂ψ hik = 0 terms of observable quantities. So teleporting photons move ∂xi ∂xk along trajectories which are elements of the world-cone (like ∂ψ because there ω = ∗ = 0, putting the equation’s time term the light cone) in that space-time area where substantial ∂t particles may be teleported (the metric inside that area has into zero. It is a standing wave equation. So, from the view- been obtained above). point of a regular observer, in the frames of the particle- Considering the photon teleportation cone equation from wave concept, all particles located in zero-space manifest as the viewpoint of a regular observer, we can see that the spatial standing light-like waves, so that all zero-space appears filled 2 i k observable metric dσ = hik dx dx becomes degenerate, with a system of light-like standing waves — a light-like h= det h =0, in the space-time area of that cone. Taking hologram. This implies that an experiment for discovering || ik|| the relationship g = hg00 [1, 2] into account, we conclude non-quantum teleportation of photons should be linked to − 2 α β that the four-dimensional metric ds = gαβ dx dx degen- stationary light. erates there as well g = det gαβ =0. The last fact implies There is no problem in photon teleportation being realised that signature conditions defining|| || pseudo-Riemannian spaces along fully degenerate world-trajectories (g = 0) outside the are broken, so that photon teleportation manifests outside the basic pseudo-Riemannian space (g < 0), while teleportation basic space-time of the General Theory of Relativity. Such trajectories of substantial particles are strictly non-degenerate a fully degenerate space was considered in [3, 4], where it (g < 0) so the lines are located in the pseudo-Riemannian was referred to as a zero-space because, from viewpoint of space†. It presents no problem because at any point of the a regular observer, all spatial intervals and time intervals are pseudo-Riemannian space we can place a tangential space zero there. of g 6 0 consisting of the regular pseudo-Riemannian space When dτ = 0 and dσ = 0 observable relativistic mass m (g < 0) and the zero-space (g = 0) as two different areas of and the frequency ω become zero. Thus, from the viewpoint the same manifold. A space of g 6 0 is a natural general- of a regular observer, all particles located in zero-space ization of the basic space-time of the General Theory of (in particular, teleporting photons) having zero rest-masses Relativity, permitting non-quantum ways for teleportation of m0 = 0 appear as zero relativistic masses m = 0 and the fre- both photons and substantial particles (previously achieved quencies ω = 0. Therefore particles of this kind may be as- only in quantum fashion — quantum teleportation of photons sumed to be the ultimate case of massless light-like particles. in 1998 [6] and of atoms in 2004 [7, 8]). We will refer to all particles located in zero-space as Until now teleportation has had an explanation given only zero-particles. by Quantum Mechanics [9]. Now the situation changes: with In the frames of the particle-wave concept each particle our theory we can find physical conditions for the realisation ∂ψ is given by its own wave world-vector K = , where ψ of teleportation of both photons and substantial particles in α ∂xα a non-quantum way. is the wave phase (eikonal). The eikonal equation K Kα = 0 α The only difference is that from the viewpoint of a regular [5], setting forth that the length of the wave vector Kα re- observer the square of any parallely transported vector rem- mains unchanged , for regular massless light-like particles ∗ ains unchanged. It is also an “observable truth” for vectors (regular photons), becomes a travelling wave equation in zero-space, because the observer reasons standards of his 1 ∂ψ 2 ∂ψ ∂ψ pseudo-Riemannian space anyway. The eikonal equation in ∗ hik ∗ ∗ = 0 , c2 ∂t − ∂xi ∂xk zero-space, expressed in his observable world-coordinates, is α i ∂ψ ∂ψ KαK =0. But in ideal world-coordinates t and x the metric α αβ 2 that may be obtained after taking KαK = g α β = 0 2 w 2 2 i k ∂x ∂x inside zero-space, ds = 1 c dt + gik dx dx = 0, in the terms of physical observable quantities [1, 2], where we − − c2 degenerates into a three-dimensional dμ2 which, depending ∗According to Levi-Civita’s rule, in a Riemannian space of n dimen- α sions the length of any n-dimensional vector Q remains unchanged †Any space of Riemannian geometry has the strictly non-degenerate α in parallel transport, so QαQ = const. So it is also true for the four- metric feature g < 0 by definition. Pseudo-Riemannian spaces are a dimensional wave vector Kα in a four-dimensional pseudo-Riemannian particular case of Riemannian spaces, where the metric is sign-alternating. (+ ) space — the basic space-time of the General Theory of Relativity. Since So the four-dimensional pseudo-Riemannian space of the signature −−− ( +++) ds = 0 is true along isotropic trajectories (because cdτ = dσ), the length of or − on which Einstein based the General Theory of Relativity is also α any isotropic vector is zero, so that we have KαK = 0. a strictly non-degenerated metric (g < 0).
18 L. Borissova and D. Rabounski. On the Possibility of Instant Displacements in the Space-Time of General Relativity April, 2005 PROGRESS IN PHYSICS Volume 1
on gravitational potential w uncompensated by something space is a point (dτ = 0, dσ = 0), while dτ = 0 and dσ = 0 2 2 else, is not invariant, dμ2 = g dxidxk = 1 w c2dt2 = taken in ideal world-coordinates become 1 w c2dt2 + ik − c2 6 − − c2 = inv. As a result, within zero-space, the square of a trans- + g dxidxk = 0, which is a four-dimensional cone equation 6 ik ported vector, a four-dimensional coordinate velocity vector like the light cone. Actually here is the “Finite Relativity U α for instance, being degenerated into a spatial U i, does Principle” for observed objects — an observed point is the not remain unchanged whole space taken in ideal coordinates. w 2 U U i = g U iU k = 1 c2 = const, Conclusions i ik − c2 6 so that although the geometry is Riemannian for a regular This research currently is the sole explanation of virtual par- observer, the real geometry of zero-space within the space ticles and virtual interaction given by the purely geometrical itself is non-Riemannian. methods of Einstein’s theory. It is possible that this method We conclude from this brief study that instant displace- will establish a link between Quantum Electrodynamics and ments of particles are naturally permitted in the space-time the General Theory of Relativity. of the General Theory of Relativity. As it was shown, tele- Moreover, this research is currently the sole theoretical portation of substantial particles and photons realizes itself in explanation of the observed phenomenon of teleportation different space-time areas. But it would be a mistake to think [6, 7, 8] given by the General Theory of Relativity. that teleportation requires the acceleration of a substantial This paper was read at the conference “Today’s Take on particle to super-light speeds (the tachyons area), while a Einstein’s Relativity” (Tucson, Arizona), February 18, 2005. photon needs to be accelerated to infinite speed. No — as it The authors are grateful to Prof. Florentin Smarandache for i 2 his assistance and Stephen J. Crothers for some editing. is easy to see from the teleportation condition w + viu = c , if gravitational potential is essential and the space rotates at a speed close to the velocity of light, substantial particles References may be teleported at regular sub-light speeds. Photons can reach the teleportation condition easier, because they move 1. Zelmanov A. L. Chronometric invariants. Dissertation, 1944. First published: CERN, EXT-2004-117, 236 pages. at the velocity of light. From the viewpoint of a regular observer, as soon as the teleportation condition is realised 2. Zelmanov A. L. Chronometric invariants and co-moving coordinates in the general relativity theory. Doklady Acad. in the neighbourhood of a moving particle, such particle Nauk USSR, 1956, v. 107 (6), 815–818. “disappears” although it continues its motion at a sub-light 3. Rabounski D. D. and Borissova L. B. Particles here and beyond coordinate velocity ui (or at the velocity of light) in another the Mirror. Editorial URSS, Moscow, 2001, 84 pages. space-time area invisible to us. Then, having its velocity reduced, or by the breaking of the teleportation condition by 4. Borissova L. B. and Rabounski D. D. Fields, vacuum, and the mirror Universe. Editorial URSS, Moscow, 2001, 272 pages something else (lowering gravitational potential or the space (the 2nd revised ed.: CERN, EXT-2003-025). rotation speed), it “appears” at the same observable moment 5. Landau L. D. and Lifshitz E. M. The classical theory of fields. at another point of our observable space at that distance and i GITTL, Moscow, 1939 (ref. with the 4th final exp. edition, in the direction which it obtained by u there. Butterworth–Heinemann, 1980). In connection with the results, it is important to re- 6. Boschi D., Branca S., De Martini F., Hardy L., and Popescu S. member the “Infinity Relativity Principle”, introduced by Experimental realization of teleporting an unknown pure Abraham Zelmanov (1913–1987), a prominent cosmologist. quantum state via dual classical and Einstein-Podolsky-Rosen Proceeding from his cosmological studies [1], he concluded Channels. Phys. Rev. Lett., 1998, v. 80, 1121–1125. that “. . . in homogeneous isotropic cosmological models spa- 7. Riebe M., Haffner¨ H., Roos C. F., Hansel¨ W., Benhelm J., La- tial infinity of the Universe depends on our choice ofthat ncaster G. P.T., Korber T. W., Becher C., Schmidt-Kaler F., reference frame from which we observe the Universe (the James D. F. V., and Blatt R. Deterministic quantum telepor- observer’s reference frame). If the three-dimensional space tation with atoms. Nature, 2004, v. 429, 734–736. of the Universe, being observed in one reference frame, is 8. Barrett M. D., Chiaverini J., Schaetz T., Britton J., Itano W. M., infinite, it may be finite in another reference frame. The Jost J. D., Knill E., Langer C., Leibfried D., Ozeri R., same is as well true for the time during which the Universe Wineland D. J. Deterministic quantum teleportation of atomic evolves.” qubits. Nature, 2004, v. 429, 737–739. We have come to the “Finite Relativity Principle” here. 9. Bennett C. H., Brassard G., Crepeau C., Jozsa R., Peres A., As it was shown, because of a difference between phys- and Wootters W. K. Teleporting an unknown quantum state ical observable world-coordinates and ideal ones, the same via dual classical and Einstein-Podolsky-Rosen channels. Phys. space-time areas may be very different, being defined in each Rev. Lett., 1993, v. 70, 1895–1899. of the frames. Thus, in observable world-coordinates, zero-
L. Borissova and D. Rabounski. On the Possibility of Instant Displacements in the Space-Time of General Relativity 19 Volume 1 PROGRESS IN PHYSICS April, 2005
On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics
Carlos Castro Center for Theor. Studies of Physical Systems, Clark Atlanta University, Atlanta, USA E-mail: [email protected]; [email protected]
We investigate the consequences of the Mach’s principle of inertia within the context of the Dual Phase Space Relativity which is compatible with the Eddington-Dirac large numbers coincidences and may provide with a physical reason behind the observed anomalous Pioneer acceleration and a solution to the riddle of the cosmological constant problem. The cosmological implications of Non-Archimedean Geometry by assigning an upper impossible scale in Nature and the cosmological variations of the fundamental constants are also discussed. We study the corrections to Newtonian dynamics resulting from the Dual Phase Space Relativity by analyzing the behavior of a test particle in a modified Schwarzschild geometry (due to the the effects of the maximal acceleration) that leads in the weak-field approximation to essential modifications of the Newtonian dynamics and to violations of the equivalence principle. Finally we follow another avenue and find modified Newtonian dynamics induced by the Yang’s Noncommutative Spacetime algebra involving a lower and upper scale in Nature.
1 Introduction The idea of minimal length (the Planck scale LP ) can be incorporated within the context of the maximal acceleration 2 In recent years we have argued that the underlying funda- Relativity principle [68] amax = c /LP in Finsler Geom- mental physical principle behind string theory, not unlike the etries [56] and [14]. A different approach than the one based principle of equivalence and general covariance in Einstein’s on Finsler Geometries is the pseudo-complex Lorentz group general relativity, might well be related to the existence of description by Schuller [61] related to the effects of maximal an invariant minimal length scale (Planck scale) attainable acceleration in Born-Infeld models that also maintains Lo- in nature. A scale relativistic theory involving spacetime rentz invariance, in contrast to the approaches of Double resolutions was developed long ago by Nottale where the Special Relativity (DSR) [70] where the Lorentz symmetry Planck scale was postulated as the minimum observer in- is deformed. Quantum group deformations of the Poincare´ dependent invariant resolution in Nature [2]. Since “points” symmetry and of Gravity have been analyzed by [69] where cannot be observed physically with an ultimate resolution, the deformation parameter q could be interpreted in terms they are fuzzy and smeared out into fuzzy balls of Planck of an upper and lower scale as q = eLP /R such that the radius of arbitrary dimension. For this reason one must con- undeformed limit q = 1 can be attained when LP 0 and/or struct a theory that includes all dimensions (and signatures) when R [68]. For a discussions on the open problems→ of on the equal footing. Because the notion of dimension is a Double Special→ ∞ Relativity theories based on kappa-deformed topological invariant, and the concept of a fixed dimension Poincare´ symmetries [63] and motivated by the anomalous is lost due to the fuzzy nature of points, dimensions are Lorentz-violating dispersion relations in the ultra high energy resolution-dependent, one must also include a theory with cosmic rays [71, 73], we refer to [70]. all topologies as well. It turned out that Clifford algebras An upper limit on the maximal acceleration of particles contained the appropriate algebro-geometric features to im- was proposed long ago by Caianiello [52]. This idea is a plement this principle of polydimensional transformations direct consequence of a suggestion made years earlier by that reshuffle a five-brane history for a membrane history, for Max Born on a Dual Relativity principle operating in Phase example. For an extensive review of this Extended Relativity Spaces [49], [74] where there is an upper bound on the Theory in Clifford Spaces that encompasses the unified dy- four-force (maximal string tension or tidal forces in strings) namics of all p-branes, for different values of the dimensions acting on a particle as well as an upper bound in the particle’s of the extended objects, and numerous physical conse- velocity given by the speed of light. For a recent status of the quences, see [1]. geometries behind maximal-acceleration see [73]; its relation A Clifford-space dynamical derivation of the stringy- to the Double Special Relativity programs was studied by minimal length uncertainty relations [11] was furnished in [55] and the possibility that Moyal deformations of Poincare´ [45]. The dynamical consequences of the minimal-length in algebras could be related to the kappa-deformed Poincare´ Newtonian dynamics have been recently reviewed by [44]. algebras was raised in [68]. A thorough study of Finsler
20 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
geometry and Clifford algebras has been undertaken by Va- violations of the equivalence principle. For violations of the caru [81] where Clifford/spinor structures were defined with equivalence principle in neutrino oscillations see [42], [54]. respect to Nonlinear connections associated with certain non- Finally, in 4 we study another interesting avenue for the holonomic modifications of Riemann-Cartan gravity. The origins of modified Newtonian dynamics based on Yang’s study of non-holonomic Clifford-Structures in the construc- Noncommutative Spacetime algebra involving a lower and tion of a Noncommutative Riemann-Finsler Geometry has upper scale [136] that has been revisited recently by us [134] recently been advanced by [81]. in the context of holography and area-quantization in C- Other implications of the maximal acceleration principle spaces (Clifford spaces); in the physics of D-branes and in Nature, like neutrino oscillations and other phenomena, covariant Matrix models by [137] and within the context of have been studied by [54], [67], [22]. Recently, the variations Lie algebra stability by [48]. A different algebra with two of the fine structure constant α [64] with the cosmological length scales has been studied by [43] in order to account for accelerated expansion of the Universe was recast as a re- modifications of Newtonian dynamics (that also violates the normalization group-like equation governing the cosmolo- equivalence principle). gical red shift (Universe scale) variations of α based on this maximal acceleration principle in Nature [68]. The fine 2 Dual Phase-Space Relativity structure constant was smaller in the past. Pushing the cutoff scale to the minimum Planck scale led to the intriguing result In this section we will review in detail the Born’s Dual Phase that the fine structure constant could have been extremely Space Relativity and the principle of Maximal-acceleration small (zero) in the early Universe and that all matter in Relativity [68] from the perspective of 8D Phase Spaces and the Universe could have emerged via the Unruh-Rindler- the role of the invariance U(1, 3) Group. We will focus for Hawking effect (creation of radiation/matter) due to the ac- simplicity on a flat 8D Phase Space. A curved case scenario celeration w. r. t the vacuum frame of reference. For reviews has been analyzed by Brandt [56] within the context of the on the alleged variations of the fundamental constants in Finsler geometry of the 8D tangent bundle of spacetime Nature see [65]. and written the generalized 8D gravitational equations that The outline of this work goes as follows. In section 2 we reduce to the ordinary Einstein-Riemannian gravitational review the Dual Phase Space Relativity and show why the equations in the infinite acceleration limit. Vacaru [81] has Planck areas are invariant under acceleration-boosts trans- constructed the Riemann-Finsler geometries endowed with formations. non-holonomic structures induced by nonlinear connections In 3.1 we investigate the consequences of the Mach’s and developed the formalism to build a Noncommutative principle of inertia within the context of the Dual Phase Space Riemann-Finsler Geometry by introducing suitable Clifford Relativity Principle which is compatible with the Eddington- structures. A curved momentum space geometry was studied Dirac large numbers coincidence and may provide with a by [50]. Toller [73] has explored the different possible geom- very plausible physical reason behind the observed anoma- etries associated with the maximal acceleration principle and lous Pioneer acceleration due to the fact that the universe the physical implications of the meaning of an “observer”, is in accelerated motion (a non-inertial frame of reference) “measuring device” in the cotangent space. w. r. t the vacuum. Our proposal shares similarities with the The U(1, 3) = SU(1, 3) U(1) Group transformations, previous work of [6], [3]. To our knowledge, the first person which leave invariant the phase-space⊗ intervals under rota- who predicted the Pioneer anomaly in 1978 was P.LaViolette tions, velocity and acceleration boosts, were found by Low [5], from an entirely different approach based on the novel [74] and can be simplified drastically when the velocity/ac- theory of sub-quantum kinetics to explain the vacuum fluctu- celeration boosts are taken to lie in the z-direction, leaving ations, two years prior to the Anderson et al observations [7]. the transverse directions x, y, px, py intact; i. e., the U(1, 1) = The cosmological implications of Non-Archimedean Geom- = SU(1, 1) U(1) subgroup transformations that leave in- etry [94] by assigning an upper impassible scale in Nature [2] variant the⊗ phase-space interval are given by (in units of and the cosmological variations of the fundamental constants ~ = c = 1) are also discussed. (dE)2 (dP )2 In 3.2 the crucial modifications to Newtonian dynamics (dω)2 = (dT )2 (dX)2 + − = resulting from the Dual Phase Space Relativity are analyzed − b2 further. In particular, the physical consequences of an upper (dE/dτ)2 (dP/dτ)2 = (dτ)2 1 + − = (2.1) and lower bounds in the acceleration and an upper and b2 lower bounds in the angular velocity. We study the particular m2g2(τ) behavior of a test particle living in a modified Schwarzschild = (dτ)2 1 , − m2 A2 geometry (due to the the effects of the principle of maximal P max acceleration) that leads in the weak-field approximation to where we have factored out the proper time infinitesimal essential modifications of the Newtonian dynamics and to (dτ)2 = dT 2 dX2 in eq-(2.1) and the maximal proper- −
C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics 21 Volume 1 PROGRESS IN PHYSICS April, 2005
force is set to be b mP Amax. Here mP is the Planck mass Straightforward algebra allows us to verify that these ≡ 2 1/LP so that b = (1/LP ) , may also be interpreted as the transformations leave the interval of eq-(2.1) in classical maximal string tension when LP is the Planck scale. phase space invariant. They are are fully consistent with The quantity g(τ) is the proper four-acceleration of a Born’s duality Relativity symmetry principle [49] (Q, P ) particle of mass m in the z-direction which we take to be (P, Q). By inspection we can see that under Born du-→ defined by the X coordinate. The interval (dω)2 described ality,→ the− transformations in eqs-(2.2a, 2.2d) are rotated into by Low [74] is U(1, 3)-invariant for the most general trans- each other, up to numerical b factors in order to match formations in the 8D phase-space. These transformations units. When on sets ξa = 0 in (2.2a, 2.2d) one recovers are rather elaborate, so we refer to the references [74] for automatically the standard Lorentz transformations for the details. The appearance of the U(1, 3) group in 8D Phase X,T and E,P variables separately, leaving invariant the Space is not too surprising since it could be seen as the intervals dT 2 dX2 = (dτ)2 and (dE2 dP 2)/b2 sepa- “complex doubling” version of the Lorentz group SO(1, 3). rately. − − Low discussed the irreducible unitary representations of such When one sets ξv = 0 we obtain the transformations rules U(1, 3) group and the relevance for the strong interactions of the events in Phase space, from one reference-frame into of quarks and hadrons since U(1, 3), with 16 generators, another uniformly-accelerated frame of reference, a = const, contains the SU(3) group. whose acceleration-rapidity parameter is in this particular The analog of the Lorentz relativistic factor in eq-(2.1) case: ξ ma involves the ratios of two proper forces. One variable force is ξ a , tanh(ξ) = . (2.5) given by mg(τ) and the maximal proper force sustained by ≡ b mP Amax an elementary particle of mass mP (a Planckton) is assumed The transformations for pure acceleration-boosts in Phase 2 to be Fmax = mP lanckc /LP . When m = mP , the ratio- Space are: squared of the forces appearing in the relativistic factor of P 2 2 T 0 = T cosh ξ + sinh ξ , (2.6a) eq-(2.1) becomes then g /Amax, and the phase space interval b coincides with the geometric interval discussed by [61], [54], E0 = E cosh ξ bX sinh ξ , (2.6b) [67], [22]. − E The transformations laws of the coordinates in that leave X0 = X cosh ξ sinh ξ , (2.6c) invariant the interval (2.1) were given by [74]: − b P 0 = P cosh ξ + bT sinh ξ . (2.6d) ξvX ξaP sinh ξ T 0 = T cosh ξ + + , (2.2a) c2 b2 ξ It is straightforward to verify that the transformations (2.6a, 2.6c) leave invariant the fully phase space interval sinh ξ (2.1) but does not leave invariant the proper time interval E0 = E cosh ξ + ( ξ X + ξ P ) , (2.2b) − a v ξ (dτ)2 = dT 2 dX2. Only the combination: − ξaE sinh ξ 2 2 X = X cosh ξ + ξ T , (2.2c) 2 2 m g 0 v 2 (dω) = (dτ) 1 (2.7a) − b ξ − m2 A2 P max ξvE sinh ξ P 0 = P cosh ξ + + ξ T . (2.2d) is truly left invariant under pure acceleration-boosts in Phase c2 a ξ Space. Once again, can verify as well that these transforma- The ξv is velocity-boost rapidity parameter and the ξa is tions satisfy Born’s duality symmetry principle: the force/acceleration-boost rapidity parameter of the primed-reference frame. They are defined respectively: (T,X) (E,P ), (E,P ) ( T, X) (2.7b) → → − − ξ v ξ ma and b 1 . The latter Born duality transformation is nothing tanh v = , tanh a = . (2.3) → b c c b mP Amax but a manifestation of the large/small tension duality princi- ple reminiscent of the T -duality symmetry in string theory; The effective boost parameter ξ of the U(1, 1) subgroup i. e. namely, a small/large radius duality, a winding modes/ transformations appearing in eqs-(2.2a, 2.2d) is defined in Kaluza-Klein modes duality symmetry in string compacti- terms of the velocity and acceleration boosts parameters fications and the Ultraviolet/Infrared entanglement in Non- ξv, ξa respectively as: commutative Field Theories. Hence, Born’s duality principle in exchanging coordinates for momenta could be the under- ξ2 ξ2 ξ v + a . (2.4) lying physical reason behind T -duality in string theory. ≡ c2 b2 r The composition of two successive pure acceleration- Our definition of the rapidity parameters are different boosts is another pure acceleration-boost with acceleration than those in [74]. rapidity given by ξ00 = ξ + ξ0. The addition of proper
22 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
forces (accelerations) follows the usual relativistic compo- 1 ΔE0 = (cosh ξ sinh ξ) 0 (2.11b) sition rule: LP − → by a simple use of L’Hopital’sˆ rule or by noticing that both tanh ξ00 = tanh(ξ + ξ0) = cosh ξ; sinh ξ functions approach infinity at the same rate ma + ma0 tanh ξ + tanh ξ0 ma00 mP A mP A (2.8) ΔX0 = L (cosh ξ sinh ξ) 0 , (2.11c) = = m2aa P 1 + tanh ξ tanh ξ ⇒ mP A 0 − → 0 1 + m2 A2 P 1 ΔP 0 = (cosh ξ + sinh ξ) , (2.11d) and in this fashion the upper limiting proper acceleration is LP → ∞ never surpassed like it happens with the ordinary Special where the discrete displacements of two events in Phase Spa- 1 Relativistic addition of velocities. ce are defined: ΔX = X2 X1 = LP , ΔE = E2 E1 = , − − LP The group properties of the full combination of velocity ΔT = T T = L and ΔP = P P = 1 . 2 1 P 2 1 LP and acceleration boosts eqs-(2.2a, 2.2d) in Phase Space re- Due to− the identity: − quires much more algebra [68]. A careful study reveals that the composition rule of two successive full transformations (cosh ξ + sinh ξ)(cosh ξ sinh ξ) = − 2 2 (2.12) is given by ξ00 = ξ + ξ0 and the transformation laws are pre- = cosh ξ sinh ξ = 1 served if, and only if, the ξ; ξ0; ξ00 ... parameters obeyed the − suitable relations: one can see from eqs-(2.11a, 2.11d) that the Planck-scale Areas are truly invariant under infinite acceleration-boosts ξa ξa0 ξa00 ξa00 ξ = : = = = , (2.9a) ∞ ξ ξ0 ξ00 ξ + ξ0 ΔX0ΔP 0 = 0 = × ∞ ξ ξ ξ ξ 2 2 LP (2.13a) v = v0 = v00 = v00 . (2.9b) =ΔXΔP (cosh ξ sinh ξ)=ΔXΔP = =1 , ξ ξ0 ξ00 ξ + ξ0 − LP Finally we arrive at the composition law for the effective, ΔT 0ΔE0 = 0 = ∞ × velocity and acceleration boosts parameters ; ; re- 2 2 LP (2.13b) ξ00 ξv00 ξa00 =ΔT ΔE (cosh ξ sinh ξ)=ΔT ΔE= =1 , spectively: − LP ξv00 = ξv + ξv0 , (2.10a) ΔX0ΔT 0 = 0 = × ∞ (2.13c) ξa00 = ξa + ξa0 , (2.10b) 2 2 2 =ΔXΔT (cosh ξ sinh ξ)=ΔXΔT =(LP ) , ξ00 = ξ + ξ0. (2.10c) − ΔP 0ΔE0 = 0 = The above relations among the parameters are required ∞ × (2.13d) in order to prove the U(1, 1) group composition law of the 2 2 1 = ΔP ΔE (cosh ξ sinh ξ) = ΔP ΔE = L2 . transformations in order to have a truly Maximal-Acceleration − P Phase Space Relativity theory resulting from a Phase-Space It is important to emphasize that the invariance property change of coordinates in the cotangent bundle of spacetime. of the minimal Planck-scale Areas (maximal Tension) is not an exclusive property of infinite acceleration boosts ξ = , but, as a result of the identity cosh2 ξ sinh2 ξ = 1,∞ for 2.1 Planck-scale Areas are invariant under acceleration all values of ξ, the minimal Planck-scale−Areas are always boosts invariant under any acceleration-boosts transformations. Having displayed explicitly the Group transformations rules Meaning physically, in units of ~ = c = 1, that the Maximal 1 of the coordinates in Phase space we will show why infinite Tension (or maximal Force) b = 2 is a true physical invar- LP acceleration-boosts (which is not the same as infinite proper iant universal quantity. Also we notice that the Phase-space acceleration) preserve Planck-scale Areas [68] as a result of areas, or cells, in units of ~, are also invariant! The pure- 2 the fact that b = (1/LP ) equals the maximal invariant force, acceleration boosts transformations are “symplectic”. It can or string tension, if the units of ~ = c = 1 are used. be shown also that areas greater (smaller) than the Planck- At Planck-scale LP intervals/increments in one reference area remain greater (smaller) than the invariant Planck-area frame we have by definition (in units of ~ = c = 1): ΔX = under acceleration-boosts transformations. 1 1 = ΔT = LP and ΔE = ΔP = L where b L2 is the max- The infinite acceleration-boosts are closely related tothe P ≡ P imal tension. From eqs-(2.6a, 2.6d) we get for the trans- infinite red-shift effects when light signals barely escape formation rules of the finite intervals ΔX, ΔT , ΔE, ΔP , Black hole Horizons reaching an asymptotic observer with an from one reference frame into another frame, in the infinite infinite red shift factor. The important fact is that the Planck- acceleration-boost limit ξ , scale Areas are truly maintained invariant under acceleration- → ∞ boosts. This could reveal very important information about ΔT 0 = L (cosh ξ + sinh ξ) Black-holes Entropy and Holography. P → ∞
C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics 23 Volume 1 PROGRESS IN PHYSICS April, 2005
3 Modified Newtonian Dynamics from Phase Space c2/R is given in terms of an infrared-cutoff R (the Hubble Relativity horizon radius). Equating these proper-four forces gives
2 2 3.1 The Machian Principle and Eddington-Dirac Large mP c MU c MU R = = 1061 , (3.5) Numbers Coincidence LP R ⇒ mP LP ∼ A natural action associated with the invariant interval in from this equality of proper-four forces associated with a 61 Phase-Space given by eq-(2.1) is: maximal/minimal acceleration one infers MU 10 mP lanck 61 19 80 ∼ 10 10 mproton = 10 mproton which is indeed consis- m2 tent∼ with observations and agrees with the Eddington-Dirac S = m dτ 1 + (d2xμ/dτ 2)(d2x /dτ 2) . (3.1) m2 a2 μ number 1080: Z s P F 2 R 2 The proper-acceleration is orthogonal to the proper- N = 1080 = (1040)2 e , (3.6) velocity and this can be easily verified by differentiating ∼ F ∼ r G e the time-like proper-velocity squared: 2 2 where Fe = e /r is the electrostatic force between an elec- μ 2 2 dx dxμ μ tron and a proton; FG = Gmemproton/r is the correspond- V = = V Vμ = 1 > 0 2 13 dτ dτ ⇒ ing gravitational force and r = e /m 10− cm is the (3.2) e e ∼ dV μ d2xμ classical electron radius (in units ~ = c = 1). Vμ = Vμ = 0 , One may notice that the above equation (3.5) is also ⇒ dτ dτ 2 consistent with the Machian postulate that the rest mass of a which implies that the proper-acceleration is space-like: particle is determined via the gravitational potential energy due to the other masses in the universe. In particular, by d2xμ d2x g2(τ) = μ < 0 equating: − dτ 2 dτ 2 ⇒ (3.3) 2 2 2 2 mj GmiMU c MU m g mic =Gmi = = . (3.7) S = m dτ 1 = m dω , ri rj R ⇒ G R ⇒ − m2 a2 j Z s P Z X | − | where the analog of the Lorentz time-dilation factor in Phase- Due to the negative binding energy, the composite mass space is now given by m12 of a system of two objects of mass m1, m2 is not equal to the sum m1 + m2 > m12. We can now arrive at 2 m2g2(τ) the conclusion that the minimal acceleration c /R is also dω = dτ 1 2 2 , (3.4a) the same acceleration induced on a test particle of mass s − mP a m by a spherical mass distribution MU inside a radius R. namely, The acceleration felt by a test particle of mass m sitting at the edge of the observable Universe (at the Hubble horizon 2 2 2 2 2 m g (τ) μ ν radius R) is: (dω) = Ω dτ = 1 gμν dx dx . (3.4b) 2 2 GMU − mP a = a . (3.8) R2 The invariant proper interval is no longer the standard From the last two equations (3.7, 3.8) one gets the same proper-time τ but is given by the quantity ω(τ). The deep connection between the physics of maximal acceleration and expression for the minimal acceleration: Finsler geometry has been analyzed by [56]. The action is c2 real-valued if, and only if, m2g2 < m2 a2 in the same fashion a = a = , (3.9) P min R that the action in Minkowski spacetime is real-valued if, and only if, v2 < c2. This is the physical reason why there is an which is of the same order of magnitude as the anomalous 8 upper bound in the proper-four force acting on a fundamental acceleration of the Pioneer and Galileo spacecrafts a 10− 2 2 2 ∼ particle given by (mg)bound = mP (c /LP ) = mP in natural cm/s . A very plausible physical reason behind the observed units of ~ = c = 1. anomalous Pioneer acceleration could be due to the fact that The Eddington-Dirac large numbers coincidence (and an the universe is in accelerated expansion and motion (a non- ultraviolet/infrared entanglement) can be easily implemented inertial frame of reference) w. r. t the vacuum. Our proposal if one equates the upper bound on the proper-four force sus- shares some similarities with the previous work of [6]. To 2 tained by a fundamental particle, (mg)bound = mP (c /LP ), our knowledge, the first person who predicted the Pioneer with the proper-four force associated with the mass of the anomaly in 1978 was P.LaViolette [5], from an entirely (observed) universe MU , and whose minimal acceleration different approach based on the novel theory of sub-quantum
24 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
kinetics to explain the vacuum fluctuations, two years prior maximal-acceleration relativity principle in phase-space by to the Anderson et al observations [7]. Nottale has invoked modifying the Robertson-Friedmann-Walker metric by a the Machian principle of inertia [3] adopting a local and similar (acceleration-dependent) conformal factor as in eqs- global inertial coordinate system at the scale of the solar (3.4). It led us to the conclusion that the universe could have system in order to explain the origins of this Pioneer-Galileo emerged from the vacuum as a quantum bubble (or “brane- anomalous constant acceleration. The Dirac-Eddington large world”) of Planck mass and Planck radius that expanded number coincidences from vacuum fluctuations was studied (w. r. t to the vacuum) at the speed of light with a maximal 2 by [8]. acceleration a = c /Lp. Afterwards the acceleration began to Let us examine closer the equality between the proper- slow down as matter was being created from the vacuum, via four forces an Unruh-Rindler-Hawking effect, from this initial maximal 2 2 8 2 value c /Lp to the value of c /R 10− cm/s (of the same m c2 M c2 m M c2 ∼ P = U P = U = . (3.10) order of magnitude as the Pioneer anomalous acceleration). LP R ⇒ LP R G Namely, as the universe expanded, matter was being created from the vacuum via the Unruh-Rindler-Hawking effect The last term in eq-(3.10) is directly obtained after im- (which must not to be confused with Hoyle’s Steady State plementing the Machian principle in eq-(3.7). Thus, one Cosmolgy) such that the observable mass of the universe concludes from eq-(3.10) that as the universe evolves in enclosed within the observed Hubble horizon radius obeys time one must have the conserved ratio of the quantities (at any time) the relation M R. Such latter relationship is 2 . This interesting possibility, ad- U MU /R = c /G = mP /LP very similar (up to a factor of∼2) to the Schwarzschild black- vocated by Dirac long ago, for the fundamental constants ~, hole horizon-radius relation rs = 2M (in units of ~ = c = , , to vary over cosmological time is a plausible idea c G ... = G = 1). As matter is being created out of the vacuum, the with the provision that the above ratios satisfy the relations Hubble horizon radius grows accordingly such that M /R = in eq-(3.10) at any given moment of cosmological time. If U = c2/G. Note that the Hubble horizon radius is one-half the the fundamental constants do not vary over time then the 2 2 Schwarzchild horizon radius (1/2)(2GMU /c ) = (1/2)RS. ratio MU /R = c /G must refer then to the asymptotic values Lemaˆıtre’s idea of the Universe as a “primordial atom” of the Hubble horizon radius R = Rasymptotic. A related (like a brane-world) of Planck size has been also analyzed by approach to the idea of an impassible upper asymptotic [30] from a very different perspective than Born’s Dual Phase length R has been advocated by Scale Relativity [2] and in Space Relativity. These authors have argued that one can Khare [94] where a Cosmology based on non-Archimedean have a compatible picture of the expansion of the Universe geometry was proposed by recurring to p-adic numbers. For with the Eddington-Dirac large number coincidences if one example, a Non-Archimedean number addition law of two invokes a variation of the fundamental constants with the masses m1, m2 does not follow the naive addition rule cosmological evolution time as Dirac advocated long ago. m1 + m2 but instead: One of the most salient features of this section is that m + m it agrees with the findings of [4] where a geometric mean m m = 1 2 , 1 • 2 1 + (m m /M 2 ) relationship was found from first principles among the cos- 1 2 U 2 mological constant ρvacuum, the Planck area λ and the 2 1 2 2 which is similar to the composition law of velocities in AdS4 throat size squared R given by (ρv)− = (λ) (R ). ordinary Relativity in terms of the speed of light. When Since the throat size of de Sitter space is the same as that the masses m1, m2 are much smaller than the universe of Anti de Sitter space, by setting the infrared scale R mass MU one recovers the ordinary addition law. Similar equal to the Hubble radius horizon observed today RH and considerations follow in the Non-Archimedean composition λ equal to the Planck scale one reproduces precisely the law of lengths such that the upper length Rasym is never observed value of the vacuum energy density! [25]: ρ 2 2 4 2 122 4 ∼ surpassed. For further references on p-adic numbers and LP− lanckRH− = LP− (LP lanck/RH ) 10− MP lanck. p ∼ Physics were refer to [40]. A Mersenne prime, Mp = 2 1 = Nottale’s proposal [2] for the resolution to the cosmolo- = prime, for p = prime, p-adic hierarchy of scales in Particle− gical constant problem is based on taking the Hubble scale physics and Cosmology has been discussed by Pitkannen and R as an upper impassible scale and implementing the Scale Noyes where many of the the fundamental energy scales, Relativity principle so that in order to compare the vacuum masses and couplings in Physics has been obtained [41], energies of the Universe at the Planck scale ρ(LP ) with the 127 [42]. For example, the Mersenne prime M127 = 2 1 vacuum energy measured at the Hubble scale ρ(R) one needs 1038 (m /m )2 . The derivation of the Standard− ∼ to include the Scale Relativistic correction factors which ∼ P lanck proton Model parameters from first principle has obtained by Smith account for such apparent huge discrepancy: ρ(LP )/ρ(R) = 2 122 [43] and Beck [47]. = (R/LP ) 10 . In contrast, the results of this work are In [68] we proposed a plausible explanation of the vari- based on Born’s∼ Dual Phase-Space Relativity principle. In able fine structure constant phenomenon based onthe the next sections we will review the dynamical consequences
C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics 25 Volume 1 PROGRESS IN PHYSICS April, 2005
of the Yang’s Noncommutative spacetime algebra comprised the final expression for the conformal factor Ω2 in the case of two scales, the minimal Planck scale Lp (related to a of pure radial motion [22]: minimum distance) and an upper infrared scale R related to a Ω2(m, a, M, E, r) = minimum momentum p = ~/R. Another interesting approach 2 2 to dark matter, dark energy and the cosmological constant m 1 1 M − based on a vacuum condensate has been undertaken by [25]. = 1 2 (1 2M/r) 2 − mP a − r − We finalize this subsection by pointing out that the (3.12) 2 2 4 3 maximal/minimal angular velocity correspond to c/L and 4M (E/m) r− (1 2M/r)− P − − × c/R respectively. A maximum angular velocity has important (E/m)2 (1 2M/r) . consequences in future Thomas-precession experiments [61], × − − [73] whereas a minimal angular velocity has important con- sequences in galactic rotation measurements. The role of In the Newtonian limit, to a first order approximation, the Machian principle in constructing quantum cosmologies, we can set 1 2M/r 1 in eq-(3.12) since we shall be con- − ∼ models of dark energy, etc. . . has been studied in [52] and centrating in distances larger than the Schwarzschild radius 2 its relationship to modified Newtonian dynamics and fractals r > rs = 2M, the conformal factor Ω in eq-(3.12) simplifies: by [54], [3]. m 2 1 M 2 Ω2 1 ∼ − m a2 r2 − 3.2 Modified Newtonian Dynamics from Phase-Space P (3.13) Relativity 2 2 4 2 4M (E/m) r− (E/m) 1 , − − Having displayed the cosmological features behind the 2 proper-four forces equality (3.10) that relates the maximal/ the modified Schwarzschild metric component g000 =Ω g00 = 2 minimal acceleration in terms of the minimal/large scales and = Ω (1 2M/r) = 1 + 2 0 yields the modified gravitational − U which is compatible with Eddington-Dirac’s large number potential 0 in the weak field approximation coincidences we shall derive next the modified Newtonian U dynamics of a test particle which emerges from the Born’s g000 = 1 + 2 0 U ∼ Dual Phase Space Relativity principle. 2M m 2 1 2M 2 (3.14) The modified Schwarzschild metric is defined in terms 1 F (E/m) ∼ − r − m a2 r2 of the non-covariant acceleration as: P with 2 2 2 (dω) = Ω (dτ) = E 2 E 4 1 F (E/m) = + , (3.15) m2g (d2xμ/dτ 2)(d2xν /dτ 2) m − m 4 = 1 + μν g dxμdxν , m2 a2 μν P where F (E/m) > 0 in the Newtonian limit E < m. The modified radial acceleration which encodes the modified 2 2 μ 2 2 ν 2 g (τ) gμν (d x /dτ )(d x /dτ ) < 0 . (3.11a) Newtonian dynamics and which violates the equivalence − ≡ principle (since the acceleration now depends on the mass of A covariant acceleration in curved space-times is the test particle m) is given by: Dvμ d2xμ dxν dxρ ∂ M E m 2 = + Γμ . 0 2 νρ a0 = U = 2 1 + 8F dτ dτ dτ dτ − ∂r − r m mP × (3.16) A particle in free fall follows a geodesic with zero co- M 1 6 variant acceleration. Hence, we shall use the non-covariant + O(r− ) , × m m3 r3 acceleration in order to compute the effects of the maximal P P acceleration of a test particle in Schwarzschild spacetimes. this result is valid for distances r rs = 2M. We have set c2 The components of the non-covariant four-acceleration the maximal acceleration a = L = mP in units of ~ = c = 2 μ 2 P d x /dτ of a test particle of mass m moving in a Schwarz- = G = 1. This explains the presence of the mP factors in schild spacetime background can be obtained in a straight- the denominators. The first term in eq-(3.16) is the standard forward fashion after using the on-shell condition Newtonian gravitational acceleration M/r2 and the second μ ν 2 − 5 gμν P P = m in spherical coordinates (by solving the rela- terms are the leading corrections of order 1/r . The higher 6 tivistic Hamilton-Jacobi equations). The explict components order corrections O(r− ) appear when we do not set of the (space-like) proper-four acceleration can be found in 1 2M/r 1 in the expression for the conformal factor Ω2 [22], [36] in terms of two integration constants, the energy and− when∼ we include the extra term in the product of Ω2 E and angular momentum L. The latter components yields with g = (1 2M/r). 00 −
26 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
The conformal factor Ω2 when L = 0 (rotational degrees network sites [111] and the minimal Planck scale emerges of freedom are switched on) such that6 the test particle moves from a regularization procedure. in the radial and transverse (angular) directions has been The Yang’s algebra can be written in terms of the 6D given in [22]: angular momentum operators and a 6D pseudo-Euclidean metric ηMN : 2 m 1 μν μν 56 56 Ω2 = 1 Mˆ = ~Σ , Mˆ = ~Σ , (4.1) − m2 a2 1 2M/r × P − μ5 μ ~ μ6 μ 2 λΣ =x ˆ , Σ =p ˆ , (4.2) 3ML2 L2 M + + R × − m2r4 m2r3 − r2 λ = Σ56 , (4.3) (3.17) N m2 4L2 4E2M 2 R + + as follows: 2 2 2 4 2 4 3 ~ mP a −m r m r (1 2M/r) × [ˆpμ, ] = iη66 xˆμ , (4.4) − N − R2 E2 L2 L2 (1 2M/r) 1 + . [ˆxμ, ] = iη55 P pˆμ , (4.5) × m2 − − m2r2 N ~ [ˆxμ, xˆν ] = iη55L2 Σμν , (4.6) Following the same weak field approximation procedure − P 2 2 g000 = Ω (E, L, m)g00 = 1 + 2 0 yields the modified gravi- μ ν 66 ~ μν U [ˆp , pˆ ] = iη 2 Σ , (4.7) tational potential 0 and modified Newtonian dynamics − R U a0 = ∂r 0 that leads once again to a violation of the equiv- μ μ μν − U [ˆx , pˆ ] = i~η , (4.8) alence principle due to the fact that the acceleration depends N on the values of the masses of the test particle. [ˆxμ, Σνρ] = ημρxν ημν xρ , (4.9) − μ νρ μρ ν μν ρ 4 Modified Newtonian Dynamics resulting from Yang’s [ˆp , Σ ] = η p η p , (4.10) − Noncommutative Spacetime Algebra The dynamical consequences of the Yang’s Noncommu- tative spacetime algebra can be derived from the quantum/ We end this work with some relevant remarks about the classical correspondence: impact of Yang’s Noncommutative spacetime algebra on modified Newtonian dynamics. Such algebra involves two 1 [A,ˆ Bˆ] A, B PB , (4.11) length scales, the minimal Planck scale LP = λ and an upper i~ ↔ { } infrared cutoff scale . i. e. commutators correspond to Poisson brackets. More pre- R Recently in [134] an isomorphism between Yang’s Non- cisely, to Moyal brackets in Phase Space. In the classical limit commutative space-time algebra (involving two length sca- ~ 0 Moyal brackets reduce to Poisson brackets. Since the les) [136] and the holographic area coordinates algebra of C- coordinates→ and momenta are no longer commuting variables spaces (Clifford spaces) was constructed via an AdS5 space- the classical Newtonian dynamics is going to be modified time (embedded in 6D) which is instrumental in explaining since the symplectic two-form ωμν in Phase Space will have the origins of an extra (infrared) scale in conjunction to additional non-vanishing elements stemming from these non- R the (ultraviolet) Planck scale λ characteristic of C-spaces. commuting coordinates and momenta. Yang’s Noncommutative space-time algebra allowed Tanaka In particular, the modified brackets read now: [137] to explain the origins behind the discrete nature of the A(x, p),B(x, p) = ∂ Aωμν ∂ B = spectrum for the spatial coordinates and spatial momenta {{ }} μ ν which yields a minimum length-scale λ (ultraviolet cutoff) = A(x, p),B(x, p) xμ, pν + PB (4.12) and a minimum momentum p = ~/ (maximal length , { } { } R R ∂A ∂B ∂A ∂B infrared cutoff). + xμ, xν + pμ, pν . Related to the issue of area-quantization, the norm- ∂xμ ∂xν { } ∂pμ ∂pν { } 2 AB squared A of the holographic Area operator XABX If the coordinates and momenta were commuting vari- in Clifford-spaces has a correspondence with the quadratic ables the modified bracket will reduce to the first term only: 4 AB Casimir operator λ ΣABΣ of the conformal algebra A(x, p),B(x, p) = SO(4, 2) (SO(5, 1) in the Euclideanized AdS5 case). This {{ }} holographic area-Casimir relationship does not differ much μ ν = A(x, p),B(x, p) PB x , p = from the area-spin relation in Loop Quantum Gravity A2 { } { } (4.13) 4 2 ∼ ∂A ∂B ∂A ∂B λ ji (ji + 1) in terms of the SU(2) Casimir J with = ημν . eigenvalues j(j + 1), where the sum is taken over the spin ∂xμ ∂pν − ∂pμ ∂xν N P
C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics 27 Volume 1 PROGRESS IN PHYSICS April, 2005
The ordinary Heisenberg (canonical) algebra is recovered upper scale. The de Sitter rigid Top can be generalized when 1 in eq-(4.13). further to Clifford spaces since a Clifford-polyparticle has InN the → nonrelativistic limit, the modified dynamical equa- more degrees of freedom than a relativistic top in ordinary tions are: spacetimes [46] and, naturally, to study the modified Nambu- Poisson dynamics of p-branes [49] as well. A different phys- dxi ∂H ∂H = xi,H = xi, pj + xi, xj , (4.14) ical approach to the theory of large distance physics based on dt ∂pj ∂xj {{ }} { } { } certain two-dim nonlinear sigma models has been advanced i by Friedan [51]. dp i ∂H i j ∂H i j = p ,H = x , p + p , p . (4.15) An Extended Relativity theory with both an upper and dt {{ }} −∂xj { } ∂pj { } lower scale can be formulated in the Clifford extension of The non-relativistic Hamiltonian for a central potential Phase Spaces along similar lines as [1], [68] by adding the V (r) is: Clifford-valued polymomentum degrees of freedom to the i 1/2 pip i Clifford-valued holographic coordinates. The Planck scale H = + V (r) , r = xix . (4.16) 2m LP and the minimum momentum (~/R) are introduced to hXi i match the dimensions in the Clifford-Phase Space interval in ∂V Defining the magnitude of the central force by F = D-dimensions as follows: − ∂r and using ∂r = xi one has the modified dynamical equations ∂xi r 1 of motion (4.14, 4.15): 2 dΣ =
28 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
arXiv: gr-qc/0307042. Scale relativistic cosmology. Chaos, 22. Feoli A., Lambiase G., Papini G., and Scarpetta G. Solitons and Fractals, v. 16, 2003, 539. Schwarzschild field with maximal acceleration corections. 4. Castro C. Mod. Phys. Letters, v. A17, 2002, 2095–2103. Phys. Letters, v. A263, 1999, 147. arXiv: gr-qc/9912089. Capozziello S., Feoli A., Lambiase G., Papini G., and 5. LaViolette P. Int. Jour. Gen. Sys., v. 11, 1985, 329. Appl. J., Scarpetta G. Massive scalar particles in a Modified Schwarz- v. 301, 1986, 544. Subquantum Kinetics. Second edition, Star- schild Geometry. Phys. Letters, v. A268, 2000, 247. arXiv: lane Publishers, Alexandria, VA, 2003. gr-qc/0003087. Bozza V., Lambiase G., Papini G., and 6. Rosales J. The Pioneer’s anomalous Doppler drift as a Scarpetta G. Quantum violations of the Equivalence Principle berry phase. arXiv: gr-qc/0401014. 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Physics, v. 29, 1990, 963. Nuovo 17. Schuller F. Annals of Phys., v. 299, 2002, 174. Cimento, v. B40, 1977, 27. 18. Lukierski J., Nowicki A., Ruegg H., Tolstoy V. Phys. Letters, 32. Low S. Jour. Phys. A Math. Gen., v. 35, 2002, 5711. J. Math. v. 264, 1991, 331. Lukierski J., Ruegg H., Zakrzewski W. Phys., v. 38, 1997, 2197. Ann. Phys., v. 243, 1995, 90. Lukierski J., Nowicki A. Double 33. Vacaru S. Non-holonomic Clifford-structures and noncommu- Special Relativity verus kappa-deformed dynamics. arXiv: tative Riemann-Finsler geometry. arXiv: math.DG/0408121. hep-th/0203065. Vacaru S. and Vicol N. Nonlinear connections and Clifford 19. Webb J., Murphy M., Flambaum V., Dzuba V., Barrow J., structures. arXiv: math.DG/0205190. Vacaru S. (Non) Churchill C., Prochaska J., and Wolfe A. Further evidence commutative Finsler geometry from String/M theory. arXiv: for Cosmological Evolution of the Fine Structure Constant. hep-th/0211068. Vacaru S., Nadejda A. Int. J. Math. Sci., v. 23, Monthly Notices of the Royal Astronomical Society, v. 327, 2004, 1189–1233. Vacaru S. Clifford structures and spinors on 2001, 1208. spaces with local anisotropy. Buletinul Academiei de Stiinte 20. Uzan J. P. The fundamental constants and their variations: a Republicii Moldova, Fizica si Tehnica, (Izvestia Academii observational status and theoretical motivations. arXiv: hep- Nauk Respubliky Moldova, fizica i tehnika), v. 3, 1995, 53–62. ph/0205340. Vacaru S. Superstrings in higher order extensions of Finsler 21. Lambiase G., Papini G., Scarpetta G. Maximal acceleration superspaces. Nucl. Phys., v. B434, 1997, 590–656. corrections to the Lamb shift of one electron atoms. 34. Ashtekar A., Rovelli C., and Smolin L., Phys. Rev. Lett., v. 69, Nuovo Cimento, v. B112, 1997, 1003. arXiv: hep-th/9702130. 1992, 237. Rovelli C. A dialog on quantum gravity. arXiv: hep- Lambiase G., Papini G., Scarpetta G. Phys. Letters, v. A224, th/0310077. Freidel L., Livine E., and Rovelli C. 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35. Castro C. On noncommutative Yang’s space-time algebra, Energy. arXiv: gr-qc/0205075. Buchalter A. On the time holography, area quantization and C-space Relativity. Sub- variations of c, G and h and the dynamics of the Cosmic mitted to Class. Quan. Grav. See also CERN–EXT–2004–090. expansion. arXiv: astro-ph/0403202. 36. Yang C. N. Phys. Rev., v. 72, 1947, 874. Proc. of the Intern. 53. Roscoe D. General Relativity and Gravitation, v. 36, 2004, 3. Conf. on Elementary Particles, 1965, Kyoto, 322–323. General Relativity and Gravitation, v. 34, 2002, 577. 37. Tanaka S. Nuovo Cimento, v. 114B, 1999, 49. Tanaka S. Astrophysics and Space Science, v. 285, No. 2, 2003, 459. Noncommutative field theory on Yang’s space-time algebra, 54. Steer D., Parry M. Int. Jour. of Theor. Physics, v. 41, No. 11, covariant Moyal star products and matrix model. arXiv: hep- 2002, 2255. th/0406166. Space-time quantization and nonlocal field theory. 55. Amati D., Ciafaloni M., Veneziano G. Phys. Letters, v. B197, arXiv: hep-th/0002001. Yang’s quantized space-time algebra 1987, 81. Amati D., Ciafaloni M., Veneziano G. Phys. Letters, and holographic hypothesis. arXiv: hep-th/0303105. v. B216, 1989, 41. Gross D., Mende P. Phys. Letters, v. B197, 38. Smith F. Int. Journal Theor. Phys., v. 24, 1985, 155. Int. 1987, 129. Gross D., Mende P. Nucl. Phys., v. B303, 1988, 407. Jour. Theor. Phys., v. 25, 1985, 355. From sets to quarks. arXiv: hep-ph/9708379 and CERN–CDS: EXT–2003–087. Gonzalez-Martin G. Physical geometry. University of Simon Bolivar Publ., Caracas, 2000, 265 pages, ISBN: 9800767495. Gonzalex-Martin G. The proton/electron geometric mass ratio. arXiv: physics/0009052. Gonzalez-Martin G. The fine structure constant from relativistic groups. arXiv: physics/0009051. 39. Beck C. Spatio-Temporal Vacuum Fluctuations of Quantized Fields Advances in Nonlinear Dynamics, v. 21, World Scien- tific, Singapore 2000. Chaotic strings and standard model pa- rameters. Physica v. 171D, 2002, 72. arXiv: hep-th/0105152. 40. Vladimorov V., Volovich I., Zelenov I. P-adic Numbers in Mathematical Physics. World Scientific, Singapore, 1994. Brekke L., Freund P. Phys. Reports, v. 231, 1993, 1. 41. Pitkannen M. Topological geometrodynamics I, II. Chaos, Solitons and Fractals, v. 13, No. 6, 2002, 1205 and 1217. 42. Noyes P. Bit-String Physics: A discrete and finite approach to Natural Philosophy. Ed. by J. C. van der Berg, Series of Knots in Physics, v. 27, World Scientific, Singapore, 2001. 43. Smolin L., Kowalski-Glikman J. Triply Special Relativity. arXiv: hep-th/0406276. McGaugh S. S. Modified Newtonian Dynamics as an alternative to Dark Matter. arXiv: astro- ph/0204521. Aguirre A. Alternatives to Dark Matter. arXiv: astro-ph/0305572. 44. Nam Chang L. Some consequences of the Hypothesis of Minimal Lengths. arXiv: hep-th/0405059. 45. Castro C. Foundations of Physics, v. 8, 2000, 1301. 46. Castro C. Foundations of Physics, v. 34, No. 7, 2004, 107. 47. Armenta J., Nieto J. A. The de Sitter Relativistic Top Theory. arXiv: 0405254. 48. Vilela Mendes R. J. Phys., v. A27, 1994, 8091–8104. Vilela- Mendes R. Some consequences of a noncommutative space- time structure. arXiv: hep-th/0406013. Chryssomalakos C., Okon E. Linear form of 3-scale Special Relativity algebra and the relevance of stability. arXiv: 0407080. 49. Nambu Y. Phys. Rev., v. D7, 1973, 2405. 50. Moffat J. Quantun Gravity momentum representation an maximum invariant energy. arXiv: gr-qc/0401117. 51. Friedan D. Two talks on a tentative theory of large distance physics. arXiv: hep-th/0212268. A tentative theory of large distance physics. arXiv: hep-th/0204131. 52. Cole D. Contrasting quantum cosmologies. arXiv: gr-qc/ 0312045. Vishwakarma R. G. A Machian model of Dark
30 C. Castro. On Dual Phase-Space Relativity, the Machian Principle and Modified Newtonian Dynamics April, 2005 PROGRESS IN PHYSICS Volume 1
The Extended Relativity Theory in Clifford Spaces
Carlos Castro∗ and Matej Pavsiˇ cˇ†
∗Center for Theor. Studies of Physical Systems, Clark Atlanta University, Atlanta, USA E-mail: [email protected]; [email protected]
†Jozefˇ Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia E-mail: [email protected]
An introduction to some of the most important features of the Extended Relativity theory in Clifford-spaces (C-spaces) is presented whose “point” coordinates are non-commuting Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes. . . degrees of freedom associated with the collective particle, string, membrane, p-brane. . . dynamics of p-loops (closed p-branes) in target D-dimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, non-commuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable dimensions/signatures. It permits to study the dynamics of all (closed) p-branes, for all values of p, on a unified footing. It resolves the ordering ambiguities in QFT, the problem of time in Cosmology and admits superluminal propagation (tachyons) without violations of causality. A discussion of the maximal-acceleration Relativity principle in phase-spaces follows and the study of the invariance group of symmetry transformations in phase-space allows to show why Planck areas are invariant under acceleration-boosts transformations. This invariance feature suggests that a maximal- string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking different n-dimensional slices through C-space. The conformal group in spacetime emerges as a natural subgroup of the Clifford group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl’s gauge field of dilations. We finalize by constructing the generalization of Maxwell theory of Electrodynamics of point charges to a theory in C-spaces that involves extended charges coupled to antisymmetric tensor fields of arbitrary rank. In the concluding remarks we outline briefly the current promising research programs and their plausible connections with C-space Relativity.
Contents 5.2 C-space Klein-Gordon and Dirac wave equa- tions ...... 47 1. Introduction...... 32 6. Maximal-acceleration Relativity in phase-spaces 2. Extending Relativity from Minkowski spacetime to C-space ...... 33 6.1 Clifford algebras in phase space...... 49 6.2 Invariance under the Group ...... 50 3. Generalized dynamics of particles, fields and branes U(1, 3) in C-space 6.3 Planck-Scale Areas are invariant under accele- ration boosts ...... 52 3.1 The Polyparticle Dynamics in C-space ...... 36 3.2 A unified theory of all p-Branes in C-spaces . . . . 40 7. Some further important physical applications related to the C-space physics 4. Generalized gravitational theories in curved C-spaces: 7.1 Relativity of signature...... 52 higher derivative gravity and torsion from the geom- etry of C-space 7.2 Clifford space and the conformal group ...... 54 4.1 Ordinary space ...... 42 7.3 C-space Maxwell Electrodynamics ...... 57 4.2 C-space...... 43 8. Concluding remarks ...... 59 4.3 On the relation between the curvature of C-space and the curvature of an ordinary space ...... 46 5. On the quantization in C-spaces 5.1 The momentum constraint in C-space ...... 46
C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 31 Volume 1 PROGRESS IN PHYSICS April, 2005
1 Introduction senseless to “derive” the values of the fundamental constants in Nature from first principles, from pure thought alone; i.e. In recent years it was argued that the underlying fundamental one must invoke the Cosmological Anthropic Principle to physical principle behind string theory, not unlike the prin- explain why the constants of Nature have they values they ciple of equivalence and general covariance in Einstein’s have. general relativity, might well be related to the existence of Using these methods the bosonic p-brane propagator, an invariant minimal length scale (Planck scale) attainable in the quenched mini superspace approximation, was con- in nature [8]. A theory involving spacetime resolutions was structed in [18, 19]; the logarithmic corrections to the black developed long ago by Nottale [23] where the Planck scale hole entropy based on the geometry of Clifford space (in was postulated as the minimum observer independent invar- short C-space) were obtained in [21]; the modified nonlinear iant resolution [23] in Nature. Since “points” cannot be ob- de Broglie dispersion relations, the corresponding minimal- served physically with an ultimate resolution, it is reasonable length stringy [11] and p-brane uncertainty relations also to postulate that they are smeared out into fuzzy balls. In admitted a C-space interpretation [10], [19]. A generalization refs.[8] it was assumed that those balls have the Planck radius of Maxwell theory of electromagnetism in C-spaces com- and arbitrary dimension. For this reason it was argued in prised of extended charges coupled to antisymmetric tensor refs. [8] that one should construct a theory which includes fields of arbitrary rank was attained recently in [75]. The all dimensions (and signatures) on the equal footing. In [8] resolution of the ordering ambiguities of QFT in curved this Extended Scale Relativity principle was applied to the spaces was resolved by using polyvectors, or Clifford-algebra quantum mechanics of p-branes which led to the construction valued objects [26]. One of the most remarkable features of Clifford-space (C-space) where all p-branes were taken to of the Extended Relativity in C-spaces is that a higher de- be on the same footing, in the sense that the transformations rivative Gravity with Torsion in ordinary spacetime follows in C-space reshuffled a string history for a five-brane history, naturally from the analog of the Einstein-Hlbert action in a membrane history for a string history, for example. curved C-space [20]. Clifford algebras contained the appropriate algebraic- In this new physical theory the arena for physics is no geometric features to implement this principle of polydim- longer the ordinary spacetime, but a more general manifold of ensional transformations [14]–[17]. In [14]–[16] it was pro- Clifford algebra valued objects, noncommuting polyvectors. posed that every physical quantity is in fact a polyvector, that Such a manifold has been called a pan-dimensional con- is, a Clifford number or a Clifford aggregate. Also, spinors tinuum [14] or C-space [8]. The latter describes on a unified are the members of left or right minimal ideals of Clifford basis the objects of various dimensionality: not only points, algebra, which may provide the framework for a deeper but also closed lines, surfaces, volumes, . . . , called 0-loops understanding of sypersymmetries, i. e., the transformations (points), 1-loops (closed strings), 2-loops (closed mem- relating bosons and fermions. The Fock-Stueckelberg theory branes), 3-loops, etc. It is a sort of a dimension category, of a relativistic particle can be embedded in the Clifford where the role of functorial maps is played by C-space algebra of spacetime [15, 16]. Many important aspects of transformations which reshuffles a p-brane history for a p0- Clifford algebra are described in [1], [6], [7], [3], [15, 16, 17], brane history or a mixture of all of them, for example. The [5], [48]. It is our belief that this may lead to the proper above geometric objects may be considered as to correspond- formulation of string and M theory. ing to the well-known physical objects, namely closed p- A geometric approach to the physics of the Standard branes. Technically those transformations in C-space that Model in terms of Clifford algebras was advanced by [4]. It reshuffle objects of different dimensions are generalizations was realized in [43] that the Cl(8) Clifford algebra contains of the ordinary Lorentz transformations to C-space. the 4 fundamental nontrivial representations of Spin(8) that C-space Relativity involves a generalization of Lorentz accommodate the chiral fermions and gauge bosons of the invariance (and not a deformation of such symmetry) in- Standard Model and which also includes gravitons via the volving superpositions of p-branes (p-loops) of all possible McDowell-Mansouri-Chamseddine-West formulation of dimensions. The Planck scale is introduced as a natural para- gravity, which permits to construct locally, in D = 8, a geom- meter that allows us to bridge extended objects of different etric Lagrangian for the Standard Model plus Gravity. Fur- dimensionalities. Like the speed of light was need in Einstein thermore, discrete Clifford-algebraic methods based on Relativity to fuse space and time together in the Minkowski hyperdiamond-lattices have been instrumental in construct- spacetime interval. Another important point is that the Con- ing E8 lattices and deriving the values of the force-strengths formal Group of four-dimensional spacetime is a conse- (coupling constants) and masses of the Standard Model with quence of the Clifford algebra in four-dimensions [25] and it remarkable precision by [43]. These results have recently emphasizes the fact why the natural dilations/contractions of been corroborated by [46] for Electromagnetism, and by [47], objects in C-space is not the same physical phenomenon than where all the Standard Model parameters were obtained from what occurs in Weyl’s geometry which requires introducing, first principles, despite the contrary orthodox belief that itis by hand, a gauge field of dilations. Objects move dilationally,
32 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
in the absence of forces, for a different physical reasoning In the concluding remarks, we briefly discuss the possible than in Weyl’s geometry: they move dilationally because of avenues of future research in the construction of QFT in C- inertia. This was discussed long ago in refs. [27, 28]. spaces, Quantum Gravity, Noncommutative Geometry, and This review is organized as follows: section 2 is dedi- other lines of current promising research in the literature. cated to extending ordinary Special Relativity theory, from Minkowski spacetime to C-spaces, where the introduction 2 Extending Relativity from Minkowski spacetime to of the invariant Planck scale is required to bridge objects, C-space p-branes, of different dimensionality. The generalized dynamics of particles, fields and branes We embark into the construction of the extended relativity in C-space is studied in section 3. This formalism allows us theory in C-spaces by a natural generalization of the notion to construct for the first time, to our knowledge, a unified of a spacetime interval in Minkowski space to C-space [8, action which comprises the dynamics of all p-branes in 14, 16, 15, 17]: C-spaces, for all values of p, in one single footing (see 2 2 μ μν also [15]). In particular, the polyparticle dynamics in C- dX = dσ + dxμdx + dxμν dx + ..., (1) space, when reduced to 4-dimensional spacetime leads to the Stuckelberg formalism and the solution to the problem of where μ1 < μ2 < . . . . The Clifford valued polyvector:∗ time in Cosmology [15]. M μ μν X = X EM = σ1 + x γμ + x γμ γν + ... + In section 4 we begin by discussing the geometric Clif- ∧ (2) μ1μ2...μD ford calculus that allows us to reproduce all the standard + x γμ1 γμ2 ... γμD results in differential and projective geometry [41]. The re- ∧ ∧ solution of the ordering ambiguities of QFT in curved spaces denotes the position of a point in a manifold, called Clifford follows next when we review how it can be resolved by space or C-space. The series of terms in (2) terminates at using polyvectors, or Clifford-algebra valued objects [26]. a finite grade depending on the dimension D. A Clifford D Afterwards we construct the Generalized Gravitational The- algebra Cl(r, q) with r + q = D has 2 basis elements. For μ ories in Curved C-spaces, in particular it is shown how simplicity, the gammas γ correspond to a Clifford algebra Higher derivative Gravity with Torsion in ordinary spacetime associated with a flat spacetime: follows naturaly from the Geometry of C-space [20]. 1 In section 5 we discuss the Quantization program in γμ, γν = ημν , (3) 2 { } C-spaces, and write the C-space Klein-Gordon and Dirac equations [15]. The coresponding bosonic/fermionic p-brane but in general one could extend this formulation to curved loop-wave equations were studied by [12], [13] without em- spacetimes with metric gμν (see section 4). ploying Clifford algebra and the concept of C-space. The connection to strings and p-branes can be seen as In section 6 we review the Maximal-Acceleration Rel- follows. In the case of a closed string (a 1-loop) embedded ativity in Phase-Spaces [127], starting with the construction in a target flat spacetime background of D-dimensions, one of the submaximally-accelerated particle action of [53] using represents the projections of the closed string (1-loop) onto Clifford algebras in phase-spaces; the U(1, 3) invariance the embedding spacetime coordinate-planes by the variables transformations [74] associated with an 8-dimensional phase xμν . These variables represent the respective areas enclosed space, and show why the minimal Planck-Scale areas are by the projections of the closed string (1-loop) onto the invariant under pure acceleration boosts which suggests that corresponding embedding spacetime planes. Similary, one there could be a principle of maximal-tension (maximal can embed a closed membrane (a 2-loop) onto a D-dim flat acceleration) operating in string theory [68]. spacetime, where the projections given by the antisymmetric In section 7 we discuss the important point that the notion variables xμνρ represent the corresponding volumes enclosed of spacetime signature is relative to a chosen n-dimensional by the projections of the 2-loop along the hyperplanes of the subspace of 2n-dimensional Clifford space. Different sub- flat target spacetime background. spaces Vn — different sections through C-space — have in This procedure can be carried to all closed p-branes general different signature [15] We show afterwards how the (p-loops) where the values of p are p = 0, 1, 2, 3,... . The Conformal algebra of spacetime emerges from the Clifford p = 0 value represents the center of mass and the coordinates algebra [25] and emphasize the physical differences between xμν , xμνρ, . . . have been coined in the string-brane literature our model and the one based on Weyl geometry. At the end [24]. as the holographic areas, volumes, . . . projections of we show how Clifford algebraic methods permits one to the nested family of p-loops (closed p-branes) onto the em- generalize Maxwell theory of Electrodynamics (associated bedding spacetime coordinate planes/hyperplanes. In ref. [17] with ordinary point-charges) to a generalized Maxwell theory ∗If we do not restrict indices according to μ1 < μ2 < μ3 < . . . , then in Clifford spaces involving extended charges and p-forms the factors 1/2!, 1/3!, respectively, have to be included in front of every of arbitrary rank [75]. term in the expansion (1).
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they were interpreted as the generalized centre of mass co- shall focus solely on a finite value of D. In this fashion we ordinates of an extended object. Extended objects were thus avoid any serious algebraic convergence problems. We shall modeled in C-space. not be concerned in this work with the representations of The scalar coordinate σ entering a polyvector X is a mea- Clifford algebras in different dimensions and with different sure associated with the p-brane’s world manifold Vp+1 (e. g., signatures. the string’s 2-dimensional worldsheet V2): it is proportional The line element dS as defined in (4) is dimensionless. to the (p + 1)-dimensional area/volume of Vp+1. In other Alternatively, one can define [8, 9] the line element whose words, σ is proportional to the areal-time parameter of the dimension is that of the D-volume so that:
Eguchi-Schild formulation of string dynamics [126, 37, 24]. 2 2D 2 2D 2 μ dΣ = L dσ + L − dxμd + We see in this generalized scheme the objects as observed (5) in spacetime (which is a section through C-space) need not 2D 4 μν μ1...μD + L − dxμν dx + ... + dxμ1...μD dx . be infinitely extended along time-like directions. They need not be infinitely long world lines, world tubes. They canbe Let us use the relation finite world lines, world tubes. The σ coordinate measures γ ... γ = γ (6) how long are world lines, world tubes. During evolution they μ1 ∧ ∧ μD μ1...μD can becomes longer and longer or shorter and shorter. and write the volume element as If we take the differential dX of X and compute the scalar product among two polyvectors
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1 Since a Clifford algebra admits a matrix representation, The unitary condition Uˉ † = U − under the combined one can write the norm of a poly-vectors in terms of the reversal and complex-conjugate operation will constrain the trace operation as: X 2 = TraceX2. Hence under C-space form of the complexified boosts/rotation parameters θA ap- || || A Lorentz transformation the norms of poly-vectors behave like pearing in the rotor: U = exp θ EA . The theta parameters follows: θA are either purely real or purely imaginary depending if the reversal † , to ensure that an overall change 2 2 1 EA = EA Trace X0 = Trace [RX R− ] = ± A (15) of sign occurs in the terms θ EA inside the exponential 1 2 2 defining so that ˉ 1 holds and the norm < ˉ > = Trace [RR− X ] = Trace X . U U † = U − Z†Z s remains invariant under the analog of unitary transformations These norms are invariant under C-space Lorentz trans- in complexified C-spaces. These techniques are not very formations due to the cyclic property of the trace operation different from Penrose Twistor spaces. As far as we know 1 and RR− = 1. If one writes the invariant norm in terms a Clifford-Twistor space construction of C-spaces has not of the reversal operation
(γμ1 γμ2 ... γμp )† = γμp γμp 1 ... γμ2 γμ1 = ∧ ∧ ∧ − ∧ ∧ M M 1 M 1 M L = E RE R− E (RE R− )=E E0 , (19) p(p 1)/2 N N 0 N N = ( 1) − γ γ ... γ . h i ≡ ∗ ∗ − μ1 ∧ μ2 ∧ μp where we have labelled E as new basis element since in Hence only those terms that change sign (under the N0 the active interpretation one may perform either a change of reversal operation) are permitted in the exponential defining the polyvector components or a change of the basis elements. A . R = exp[θ EA] The means the scalar part of the expression and “ ” the Another possibility is to complexify the C-space poly- h i0 ∗ A A A scalar product. Eq-(19) has been obtained after multiplying vector valued coordinates Z = Z EA = X EA + i Y EA J J (18) from the left by E , taking into account that E EN 0 and the boosts/rotation parameters θ allowing the unitary J J h i ≡ 1 E EN = δ N , and renamiming the index J into M. condition Uˉ † = U − to hold in the generalized Clifford unit- ≡ ∗ ary transformations Z0 = UZU † associated with the com- A 3 Generalized dynamics of particles, fields and branes plexified polyvector Z = Z EA such that the interval in C-space μ μν μνρ
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description of all branes of different dimensionality. As we according to which in such a generalized dynamics an object have already indicated, since spinors are in left/right ideals may be accelerated to faster than light speeds as viewed from of a Clifford algebra, a supersymmetry is then naturally a 4-dimensional Minkowski space, which is a subspace of incorporated into this approach as well. In particular, one C-space. For a different explanation of superluminal propa- can have world manifold and target space supersymmetry gation based on the modified nonlinear de Broglie dispersion simultaneously [15]. We hope that the C-space “strings” relations see [68]. and “branes” may lead us towards discovering the physical The canonical momentum belonging to the action (20) is foundations of string and M-theory. For other alternatives κX˙ A to supersymmetry see the work by [50]. In particular, Z3 PA = . (22) ˙ B ˙ 1/2 generalizations of supersymmetry based on ternary algebras (X XB) and Clifford algebras have been proposed by Kerner [128] When the denominator in eq.-(22) is zero the momentum in what has been called Hypersymmetry. becomes infinite. We shall now calculate the speed at which this happens. This will be the maximum speed that an object 3.1 The Polyparticle Dynamics in C-space accelerating in C-space can reach. Although an initially slow object cannot accelerate beyond that speed limit, this does We will now review the theory [15, 17] in which an extended μ μν not automatically exclude the possibility that fast objects object is modeled by the components σ, x , x ,... of the traveling at a speed above that limit may exist. Such objects Clifford valued polyvector (2). By assumption the extended are C-space analog of tachyons [31, 32]. All the well known objects, as observed from Minkowski spacetime, can in gen- objections against tachyons should be reconsidered for the eral be localized not only along space-like, but also along case of C-space before we could say for sure that C-space time-like directions [15, 17]. In particular, they can be “in- tachyons do not exist as freely propagating objects. We stantonic” p-loops with either space-like or time-like orient- will leave aside this interesting possibility, and assume as ation. Or they may be long, but finite, tube-like objects. a working hypothesis that there is no tachyons in C-space. The theory that we consider here goes beyond the ordinary Vanishing of X˙ BX˙ is equivalent to vanishing of the relativity in Minkowski spacetime, therefore such localized B C-space line element objects in Minkowski spacetime pose no problems. They 0 2 1 2 01 2 are postulated to satisfy the dynamical principle which is A 2 dx dx dx dX dXA = dσ + ... formulated in C-space. All conservation laws hold in C- L − L − L2 (23) space where we have infinitely long world “lines” or Clifford dx12 2 dx123 2 dx0123 2 lines. In Minkowski spacetime M — which is a subspace of ... + + ... = 0 , 4 L2 − L3 − L4 C-space — we observe the intersections of Clifford lines with M4. And those intersections appear as localized extended where by “. . .” we mean the terms with the remaining com- objects, p-loops, described above. ponents such as x2, x01, x23,..., x012, etc. The C-space Let the motion of such an extended object be determined line element is associated with a particular choice of C- by the action principle space metric, namely GMN = EM† EN . If the basis EM , M = 1, 2,..., 2D is generated by the∗ flat space γμ satisfying ˙ ˙ 1/2 ˙ A ˙ 1/2 I = κ dτ (X† X) = κ dτ (X XA) , (20) (3), then the -space has the diagonal metric of eq.-(23) ∗ C Z Z with +, signa. In general this is not necessarily so and the where κ is a constant, playing the role of “mass” in C- C-space− metric is a more complicated expression. We take space, and τ is an arbitrary parameter. The C-space velocities now dimension of spacetime being 4, so that x0123 is the ˙ A A μ μ nu X = dX /dτ = (σ, ˙ x˙ , x˙ ,...) are also called “holo- highest grade coordinate. In eq.-(23) we introduce a length graphic” velocities. parameter L. This is necessary, since x0 = ct has dimension The equation of motion resulting from (20) is of length, x12 of length square, x123 of length to the third 0123 d X˙ A power, and x of length to the forth power. It is natural to 35 = 0 . (21) assume that L is the Planck length, that is L = 1.6 10− m. dτ ˙ B ˙ × X XB ! Let us assume that the coordinate time t = x0/c is the B A Taking X˙ X˙ B = constantp = 0 we have that X¨ = 0, so parameter with respect to which we define the speed V in that xA(τ) is a straight worldline6 in C-space. The com- C-space. ponents xA then change linearly with the parameter τ. This So we have μ means that the extended object position x , effective area dσ 2 dx1 2 dx01 2 xμν , 3-volume xμνα, 4-volume xμναβ, etc., they all change V 2 = L + + ... − dt dt L2 with time. That is, such object experiences a sort of general- 2 2 2 (24) ized dilational motion [17]. 1 dx12 1 dx123 1 dx0123 ... + + ... We shall now review the procedure exposed in ref. [17] − L dt L2 dt L3 dt −
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From eqs.-(23), (24) we find that the maximum speed is κc2 κc2 p0 = = , (27) the maximum speed is given by 2 2 1 1 dx1 1 1 dx12 c dt Lc dt V 2 = c2 . (25) − − r r which gives First, we see, the maximum speed squared V 2 contains not only the components of the 1-vector velocity dx1/dt, as it 1 dx1 1 dx12 κc2 2 = = 1 . (28) is the case in the ordinary relativity, but also the multivector c dt Lc dt s − p0 components such as dx12/dt, dx123/dt, etc. The following special cases when only certain compo- Thus to the energy of an object moving translationally at 1 nents of the velocity in C-space are different from zero, are dx /dt = 1 m/s, there corresponds the 2-vector speed d 12 d d 1 d 35 m2/s (diameter speed of particular interest: x / t = L x / t = 1.6×10− 4 × 18 ×10− m/s). This would be a typical 2-vector speed of a (i) Maximum 1-vector speed macroscopic object. For a microscopic object, such as the dx1 electron, which can be accelerated close to the speed of = c = 3.0 108m/s ; d × light, the corresponding 2-vector speed could be of the order t 26 2 13 of 10− m /s (diameter speed 10− m/s). In the examples (ii) Maximum 3-vector speed above we have provided rough estimations of possible 2- 123 vector speeds. Exact calculations should treat concrete sit- dx 2 62 3 = L c = 7.7×10− m /s ; uations of collisions of two or more objects, assume that dt not only 1-vector, but also 2-vector, 3-vector and 4-vector √3 123 d x 21 motions are possible, and take into account the conservation = 4.3×10− m/s (diameter speed) ; dt of the polyvector momentum PA. Maximum 1-vector speed, i. e., the usual speed, can ex- (iii) Maximum 4-vector speed ceed the speed of light when the holographic components 0123 12 012 dx 3 96 4 such as dσ/dt, dx /dt, dx /dt, etc., are different from = L c = 1.2 10− m /s dt × zero [17]. This can be immediately verified from eqs.-(23), √4 0123 (24). The speed of light is no longer such a strict barrier d x 24 = 1.05×10− m/s (diameter speed) . as it appears in the ordinary theory of relativity in . In dt M4 C-space a particle has extra degrees of freedom, besides the Above we have also calculated the corresponding dia- translational degrees of freedom. The scalar, σ, the bivector, meter speeds for the illustration of how fast the object ex- x12 (in general, xrs, r, s = 1, 2, 3) and the three vector, x012 pands or contracts. (in general, x0rs, r, s = 1, 2, 3), contributions to the C-space We see that the maximum multivector speeds are very quadratic form (23) have positive sign, which is just opposite small. The diameters of objects change very slowly. There- to the contributions of other components, such as xr, x0r, fore we normally do not observe the dilatational motion. xrst, xμνρσ. Because some terms in the quadratic form have Because of the positive sign in front of the σ and x12, + and some sign, the absolute value of the 3-velocity x012, etc., terms in the quadratic form (23) there are no limits dxr/dx0 can be− greater than c. to corresponding 0-vector, 2-vector and 3-vector speeds. But It is known that when tachyons can induce a breakdown if we calculate, for instance, the energy necessary to excite of causality. The simplest way to see why causality is violated 2-vector motion we find that it is very high. Or equivalently, when tachyons are used to exchange signals is by writing the to the relatively modest energies (available at the surface of temporal displacements δt = tB tA between two events (in the Earth), the corresponding 2-vector speed is very small. Minkowski space-time) in two different− frames of reference: This can be seen by calculating the energy δx 2 (δt)0=(δt) cosh(ξ)+ sinh(ξ)=(δt) cosh(ξ) + 0 κc c p = (26) (29) V 2 1 δx 1 c2 + sinh(ξ) =(δt)[cosh(ξ)+(β ) sinh(ξ)] − c δt tach. q (a) for the case of pure 1-vector motion by taking V = the boost parameter ξ is defined in terms of the velocity = dx1/dt, and as βframe = vframe/c = tanh(ξ), where vframe is is the (b) for the case of pure 2-vector motion by taking V = relative velocity (in the x-direction) of the two reference 12 = dx /(Ldt). frames and can be written in terms of the Lorentz-boost By equating the energies belonging to the cases (a) and rapidity parameter ξ by using hyperbolic functions. The Lo- 2 1/2 (b) we have rentz dilation factor is cosh(ξ) = (1 β )− ; whereas − frame
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βtachyon = vtachyon/c is the beta parameter associated with transformations. In particular, the C-space generalization of the tachyon velocity δx/δt. By emitting a tachyon along the the ordinary boost transformations associated with the boost negative x -direction one has βtachyon < 0 and such that its rapidity parameter ξ such that tanh(ξ) = βframe will involve t1 velocity exceeds the speed of light βtachyon > 1. now the family of C-space boost rapidity parameters θ , | | t12 t123 t123... A reversal in the sign of (δt)0<0 in the above boost trans- θ , θ , . . . θ , ... since boosts are just (poly) rot- formations occurs when the tachyon velocity β >1 ations along directions involving the time coordinate. Thus, | tachyon| and the relative velocity of the reference frames βframe < 1 one is replacing the ordinary boost transformations in Min- obey the inequality condition: | | kowski spacetime for the more general C-space boost trans- formations as we go from one frame of reference to another (δt)0 = (δt)[cosh(ξ) β sinh(ξ)] < 0 − | tachyon| ⇒ frame of reference. 1 1 (30) Due to the linkage among the C-space coordinates (poly- 1 < = < βtachyon ⇒ tanh(ξ) βframe | | dimensional covariance) when we envision an ordinary boost along the x1-direction, we must not forget that it is also? thereby resulting in a causality violation in the primed refer- interconnected to the area-boosts in the x12-direction as well, ence frame since the effect (event B) occurs before the cause and, which in turn, is also linked to the x2 direction. Because (event A) in the primed reference frame. the latter direction is transverse to the original tachyonic? In the case of subluminal propagation βparticle < 1 there 1-motion? the latter 2-boosts? won’t affect things and we | | x x is no causality violation since one would have: may concentrate? on the area-boosts along the x12 direction involving the θt12 parameter that will appear in the C-space (δt)0 = (δt)[cosh(ξ) β sinh(ξ)] > 0 (31) − | particle| boosts and which contribute to a crucial extra term in the due to the hyperbolic trigonometric relation: transformations such that no sign-change in δt0? will occur. More precisely, let us set all the values of the theta 2 2 t1 t12 cosh (ξ) sinh (ξ) = 1 cosh(ξ) sinh(ξ) > 0 . (32) parameters to zero except the parameters θ and θ related − ⇒ − to the ordinary boosts in the x1 direction and area-boosts in In the theory considered here, there are no tachyons in C- the x12 directions of C-space. This requires, for example, space, because physical signals in C-space are constrained that one has at least one spatial-area component, and one to live inside the C-space-light cone, defined by eq.-(23). temporal coordinate, which implies that the dimensions must However, certain worldlines in C-space, when projected onto be at least D = 2 + 1 = 3. Thus, we have in this case: the subspace M4, can appear as worldlines of ordinary tachy- 1 θt1γ γ +θt12γ γ γ ons outside the light-cone in M . The physical analog of X0 = RXR− = e t∧ 1 t∧ 1∧ 2 4 × (33) photons in C-space corresponds to tensionless p-loops, i. e., M θt1γ γ θt12γ γ γ N N M X E e− t∧ 1− t∧ 1∧ 2 X0 =L X , tensionless closed branes, since the analog of mass m in × M ⇒ M C-space is the maximal p-loop tension. By “maximal p- N N 1 where as we shown previously LM =
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transformations will be: transformations in order to check that the condition δt0 > 0 is obeyed for any physical values of the θ parameters and t t1 t12 M t t 1 (δt)0 = LM (θ ; θ )(δX ) = Lt(δt) + L1(δx ) , (35) holographic velocities. 6 According to the other author (M. P.), the problem of t 12 due to the presence of the extra term L12(δX ) in the causality could be explained as follows. In the usual theory transformations. In the more general case, when there are of relativity the existence of tachyons is problematic because more non-vanishing θ parameters, the indices M of the XM one can arrange for situations such that tachyons are sent coordinates must be restricted to those directions in C-space into the past. A tachyon T1 is emitted from an apparatus 1 12 123 0 which involve the t, x , x , x ... directions as required worldline at x1 and a second tachyon T2 can arrive to C 0 0 by the C-space poly-particle dynamics. The generalized C- the same worldline at an earlier time x0 < x and trigger C 1 space boosts involve now the ordinary tachyon velocity com- destruction of the apparatus. The spacetime event E0 at which ponent of the poly-particle as well as the generalized holo- the apparatus is destroyed coincides with the event E at graphic areas, volumes, hyper-volumes. . . velocities V M = which the apparatus by initial assumption kept on functioning M = (δX /δt) associated with the poly-vector components of normally and later emitted T1. So there is a paradox from the Clifford-valued C-space velocity. the ordinary (constrained) relativistic particle dynamics. Hence, at the expense of enlarging the ordinary Lorentz There is no paradox if one invokes the unconstrained boosts to the C-space Lorentz boosts, and the degrees of Stueckelberg description of superluminal propagation in M4. freedom of a point particle into an extended poly-particle by It can be described as follows. A C-space worldline can including the holographic coordinates, in C-space one can be described in terms of five functions xμ(τ), σ(τ) (all still have ordinary point-particle tachyons without changing other C-space coordinates being kept constant). In C-space the sign of δt, and without violating causality, due to the we have the constrained action (20), whilst in Minkowski presence of the extra terms in the C-space boosts transfor- space we have a reduced, unconstrained action. A reduction mations which ensure us that the sign of δt > 0 is maintained of variables can be done by choosing a gauge in which as we go from one frame of reference to another one. Natur- σ(τ) = τ. It was shown in ref. [16, 15, 17] that the latter ally, if one were to freeze all the θ parameters to zero except unconstrained action is equivalent to the well known Stue- one θt1 one would end up with the standard Lorentz boosts ckelberg action [33, 34]. In other words, the Stueckelberg along the x1-direction and a violation of causality would relativistic dynamics is embedded in C-space. In Stueckel- μ occur for tachyons as a result of the sign-change in δt0. berg theory all four spacetime coordinates x are independ- In future work we shall analyze in more detail if the ent dynamical degrees of freedom that evolve in terms of t M condition δt0 = LM (δX ) > 0 is satisfied for any physical an extra parameter σ which is invariant under Lorentz trans- values of the θ C-space boosts parameters and for any formations in M4. physical values of the holographic velocities consistent with From the C-space point of view, the evolution parameter M M the condition that the C-space velocity VM V > 0. What σ is just one of the C-space coordintes X . By assumption, one cannot have is a C-space tachyon; i. e. the physical σ is monotonically increasing along particles’ worldlines. signals in C-space must be constrained to live inside the Certain C-space worldlines may appear tachyonic from the C-space light-cone. The analog of “photons” in C-space point of view of M4. If we now repeat the above experiment are tensionless branes. The corresponding analog of C-space with the emission of the first and absorption of the second tachyons involve branes with imaginary tensions, not unlike tachyon we find out that the second tachyon T2 cannot ordinary tachyons m2 < 0 of imaginary mass. reach the aparatus worldline earlier than it was emmitted To sum up: Relativity in C-space demands enlarging from. Namely, T2 can arrive at a C-space event E0 with 0 0 the ordinary Lorentz group (boosts) to a larger symmetry x0 < x1, but the latter event does not coincide with the group of C-space Lorentz group and enlarging the degrees event E on the aparatus worldline, since although having μ μ of freedom by including Clifford-valued coordinates X = the same coordinates x0 = x , the events E and E0 have M = X E . This is the only way one can have a point- different extra coordinates σ0 = σ. In other words, E and E0 M 6 particle tachyonic speed in a Minkowski subspace without are different points in C-space. Therefore T2 cannot destroy violating causality in C-space. Ordinary Lorentz boosts are the apparatus and there is no paradox. incompatible with tachyons if one wishes to preserve causa- If nature indeed obeys the dynamics in Clifford space, lity. In C-space one requires to have, at least, two theta then a particle, as observed from the 4-dimensional Minkow- parameters θt1 and θt12 with the inclusion, at least, of the ski space, can be accelerated beyond the speed of light [17], t, x1, x12 coordinates in a C-space boost, to be able to provided that its extra degrees of freedom xμν , xμνα,..., μ enforce the condition δt0 > 0 under (combined) boosts along are changing simultaneously with the ordinary position x . the x1 direction accompanied by an area-boost along the x12 But such a particle, although moving faster than light in direction of C-space. It is beyond the scope of this review the subspace M4, is moving slower than light in C-space, to analyze all the further details of the full-fledged C-boosts since its speed V , defined in eq.-(24), is smaller than c. In
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this respect, our particle is not tachyon at all! In C-space tainty relations [11], [10], [19]. Thus, Quantum Mechanics we thus retain all the nice features of relativity, but in the is required to implement the postulated principle of minimal subspace M4 we have, as a particular case, the unconstrained lengths, areas, volumes. . . and which cannot be derived from Stueckelberg theory in which faster-than-light propagation the classical geometry alone. The emergence of minimal is not paradoxical and is consistent with the quantum field Planck areas occurs also in the Loop Quantum Gravity pro- theory as well [15]. This is so, because the unconstrained gram [111] where the expectation values of the Area operator Stueckelberg theory is quite different from the ordinary (con- are given by multiples of Planck area. strained) theory of relativity in M4, and faster than light Recently in [134] an isomorphism between Yang’s Non- motion in the former theory is of totally different nature from commutative space-time algebra (involving two length the faster that light motion in the latter theory. The tachyonic scales) [136] and the holographic area coordinates algebra “world lines” in M4 are just projections of trajectories in of C-spaces (Clifford spaces) was constructed via an AdS5 C-space onto Minkowski space, however, the true world space-time which is instrumental in explaining the origins of lines of M4 must be interpreted always as being embedded an extra (infrared) scale R in conjunction to the (ultraviolet) onto a larger C-space, such that they cannot take part in the Planck scale λ characteristic of C-spaces. Yang’s Noncom- paradoxical arrangement in which future could influence the mutative space-time algebra allowed Tanaka [137] to explain past. The well known objections against tachyons are not the origins behind the discrete nature of the spectrum for valid for our particle which moves according to the relativity the spatial coordinates and spatial momenta which yields a in C-space. minimum length-scale λ (ultraviolet cutoff) and a minimum We have described how one can obtain faster than light momentum p = ~/R (maximal length R, infrared cutoff). 2 motion in M4 from the theory of relativity in C-space. There In particular, the norm-squared A of the holographic Area AB are other possible ways to achieve superluminal propagation. operator XABX has a correspondence with the quadratic AB One such approach is described in refs. [84] Casimir operator ΣABΣ of the conformal algebra SO(4, 2) (SO(5, 1) in the Euclideanized AdS5 case). This An alternative procedure In ref. [9] an alternative factor- holographic area-Casimir relationship does not differ much ization of the C-space line element has been undertaken. from the area-spin relation in Loop Quantum Gravity A2 4 2 ∼ Starting from the line element dΣ of eq.-(5), instead of fac- λ ji (ji + 1) in terms of the SU (2) Casimir J with toring out the (dx0)2 element, one may factor out the (dΩ)2 eigenvalues∼ j (j + 1) and where the sum is taken over the ≡ L2Ddσ2 element, giving rise to the generalized “holo- spin networkP sites. graphic”≡ velocities measured w. r. t the Ω parameter, for ex- ample the areal-time parameter in the Eguchi-Schild formu- 3.2 A unified theory of all p-Branes in C-spaces lation of string dynamics [126], [37], [24], instead of the x0 parameter (coordinate clock). One then obtains The generalization to C-spaces of string and p-brane actions as embeddings of world-manifolds onto target spacetime μ 2 2 2D 2 dxμ dx backgrounds involves the embeddings of polyvector-valued dΣ = dΩ 1 + L − + dΩ dΩ world-manifolds (of dimensions 2d) onto polyvector-valued (36) D μν 2 target spaces (of dimensions 2 ), given by the Clifford- 2D 4 dxμν dx 2 dσ˜ + L − + ... + γ . valued maps X = X(Σ) (see [15]). These are maps from the dΩ dΩ | | dΩ Clifford-valued world-manifold, parametrized by the poly- The idea of ref. [9] was to restrict the line element (36) to vector-valued variables Σ, onto the Clifford-valued target the non tachyonic values which imposes un upper limit on the space parametrized by the polyvector-valued coordinates X. holographic velocities. The motivation was to find a lower Physically one envisions these maps as taking an n-dimen- bound of length scale. This upper holographic-velocity bound sional simplicial cell (n-loop) of the world-manifold onto an does not necessarily translate into a lower bound on the m-dimensional simplicial cell (m-loop) of the target C-space values of lengths, areas, volumes. . . without the introduction manifold; i. e. maps from n-dim objects onto m-dim objects of quantum mechanical considerations. One possibility could generalizing the old maps of taking points onto points. One is be that the upper limiting speed of light and the upper bound basically dealing with a dimension-category of objects. The size of the simplicial cells (p-loops), upon quantization of a of the momentum mpc of a Planck-mass elementary particle (the so-called Planckton in the literature) generalizes now generalized harmonic oscillator, for example, are given by to an upper-bound in the p-loop holographic velocities and multiples of the Planck scale, in area, volume, hypervolume the p-loop holographic momenta associated with elementary units or Clifford-bits. M closed p-branes whose tensions are given by powers of the In compact multi-index notation X = X ΓM one de- Planck mass. And the latter upper bounds on the holographic notes for each one of the components of the target space p-loop momenta implies a lower-bound on the holographic polyvector X: areas, volumes, . . . , resulting from the string/brane uncer- XM Xμ1μ2...μr , μ < μ < . . . < μ (37) ≡ 1 2 r
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A a b a b μν ab and for the world-manifold polyvector Σ = Σ EA: of the frame vectors e , e = eμeν g = h . Hence, the C- space world-manifold metric HAB appearing in (41) is given ΣA ξa1a2...as , a < a < . . . < a , (38) A B AB A ≡ 1 2 s by scalar product <(E )†E >0 = H , where (E )† de- notes the reversal operation of EA which requires reversing where ΓM = (1, γμ, γμν ,...) and EA = (1, ea, eab,...) form the ordering of the vectors present in the Clifford aggre- the basis of the target manifold and world manifold Clifford gate EA. algebra, respectively. It is very important to order the indices Notice, however, that the form of the action (43) is far within each multi-index M and A as shown above. The more general than the action in (39). In particular, the M A S1 above Clifford-valued coordinates X , Σ correspond to itself can be decomposed further into two additional pieces antisymmetric tensors of ranks r, s in the target spacetime by rewriting the Clifford product of two basis elements into background and in the world-manifold, respectively. a symmetric plus an antisymmetric piece, respectively: There are many different ways to construct C-space brane actions which are on-shell equivalent to the analogs of the 1 1 EAEB = EA,EB + [EA,EB] , (44) Dirac-Nambu-Goto action for extended objects and that are 2{ } 2 given by the world-volume spanned by the branes in their 1 1 motion through the target spacetime background. ΓM ΓN = ΓM , ΓN + [ΓM , ΓN ]. (45) 2{ } 2 One of these actions is the Polyakov-Howe-Tucker one: In this fashion, the S1 component has two kinds of terms. T I= [DΣ] H HAB∂ XM ∂ XNG +(2 2d) (39) The first term containing the symmetric combination is just 2 | | A B MN − the analog of the standard non-linear sigma model action, and Z p the second term is a Wess-Zumino-like term, containing the d with the 2 -dim world-manifold measure: antisymmetric combination. To extract the non-linear sigma model part of the generalized action above, we may simply [DΣ] = (dξ)(dξa)(dξa1a2 )(dξa1a2a3 ) ... (40) take the scalar product of the vielbein-variables as follows: Upon the algebraic elimination of the auxiliary world- (S1)sigma = manifold metric HAB from the action (39), via the equations T A M B N (46) of motion, yields for its on-shell solution the pullback of the = [DΣ] E <(E ∂ X Γ )†(E ∂ X Γ )> 2 | | A M B N 0 target C-space metric onto the C-space world-manifold: Z where once again we have made use of the reversal operation H (on shell) = G = ∂ XM ∂ XN G (41) AB − AB A B MN (the analog of the hermitian adjoint) before contracting multi- indices. In this fashion we recover again the Clifford-scalar upon inserting back the on-shell solutions (41) into (39) valued action given by [15]. gives the Dirac-Nambu-Goto action for the C-space branes Actions like the ones presented here in terms of deriva- directly in terms of the C-space determinant, or measure, of tives with respect to quantities with multi-indices can be the induced C-space world-manifold metric GAB, as a result mapped to actions involving higher derivatives, in the same of the embedding: fashion that the C-space scalar curvature, the analog of the Einstein-Hilbert action, could be recast as a higher derivative M N I = T [DΣ] Det(∂AX ∂BX GMN ) . (42) gravity with torsion (reviewed in sec. 4). Higher derivatives Z q actions are also related to theories of Higher spin fields [117] However in C-space, the Polyakov-Howe-Tucker action and W -geometry, W -algebras [116], [122]. For the role of admits an even further generalization that is comprised of Clifford algerbras to higher spin theories see [51]. two terms S1 + S2. The first term is [15]: The S2 (scalar) component of the C-space brane action is the usual cosmological constant term given by the C-space A B M N AB S1 = [DΣ] E E E ∂AX ∂BX ΓM ΓN . (43) determinant E = det(H ) based on the scalar part of the | | | | A B AB Z geometric product <(E )†E >0 = H Notice that this is a generalized action which is written T in terms of the C-space coordinates XM (Σ) and the C- S = [DΣ] E , (2 2d) , (47) 2 2 | | − space analog of the target-spacetime vielbein/frame one- Z m m μ M forms e = e μdx given by the Γ variables. The auxi- where the C-space determinant E = det(HAB) of the liary world-manifold vielbein variables ea, are given now by 2d 2d generalized world-manifold| | metric| HAB is given| by: the Clifford-valued frame EA variables. × p 1 In the conventional Polyakov-Howe-Tucker action, the det(HAB) = d A1A2...A2d B1B2...B2d auxiliary world-manifold metric hab associated with the stan- (2 )! × (48) A B A B A B dard p-brane actions is given by the usual scalar product H 1 1 H 2 2 ...H 2d 2d . ×
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μν μα The A1A2...A d is the totally antisymmetric tensor den- where g is the covariant metric tensor such that g gαν = 2 μ μ μ μ μ μν sity in C-space. = δ ν , γ γν + γν γ = 2δ ν and γ = g γν . There are many different forms of p-brane actions, with Following ref. [1] (see also [15]) we consider the vector and without a cosmological constant [123], and based on derivative or gradient defined according to a new integration measure by recurring to auxiliary scalar ∂ γμ∂ , (52) fields [115], that one could have used to construct their C- ≡ μ space generalizations. Since all of them are on-shell equiv- where ∂ is an operator whose action depends on the quantity alent to the Dirac-Nambu-Goto p-brane actions, we decided μ it acts on [26]. to focus solely on those actions having the Polyakov-Howe- Applying the vector derivative ∂ on a scalar field φ Tucker form. we have μ ∂φ = γ ∂μφ , (53) 4 Generalized gravitational theories in curved C-spa- where ∂ φ (∂/∂xμ)φ coincides with the partial deriva- ces: higher derivative gravity and torsion from the μ tive of φ. ≡ geometry of C-space But if we apply it on a vector field a we have
4.1 Ordinary space μ ν μ ν ν ∂a = γ ∂μ(a γν ) = γ (∂μa γν + a ∂μγν ) . (54) 4.1.1 Clifford algebra based geometric calculus in curv- ed space(time) In general γν is not constant; it satisfies the relation to works [1, 15] α Clifford algebra is a very useful tool for description of ge- ∂μγν = Γμν γα , (55) ometry, especially of curved space V . Let us first review n where Γα is the connection. Similarly, for γν = gναγ we how it works in curved space(time). Later we will discuss a μν α have generalization to curved Clifford space [20]. ∂ γν = Γν γα . (56) We would like to make those techniques accessible to μ − μα a wide audience of physicists who are not so familiar with The non commuting operator ∂μ so defined determines the rigorous underlying mathematics, and demonstrate how the parallel transport of a basis vector γν . Instead of the Clifford algebra can be straightforwardly employed in the symbol ∂μ Hestenes uses 2μ, whilst Wheeler et. al. [36] theory of gravity and its generalization. So we will leave use μ and call it “covariant derivative”. In modern, math- aside the sophisticated mathematical approach, and rather ematically∇ oriented literature more explicit notation such as follow as simple line of thought as possible, a praxis that Dγμ or γμ is used. However, such a notation, although is normally pursued by physicists. For instance, physicists mathematically∇ very relevant, would not be very practical in their works on general relativity employ a mathematical in long computations. We find it very convenient to keep formulation and notation which is much simpler from that the symbol ∂μ for components of the geometric operator μ of purely mathematical or mathematically oriented works. ∂ = γ ∂μ. When acting on a scalar field the derivative ∂μ For rigorous mathematical treatment the reader is advised to happens to be commuting and thus behaves as the ordinary study, refs. [1, 76, 77, 78, 79]. partial derivative. When acting on a vector field, ∂μ is a Let the vector fields γμ, μ = 1, 2, . . . , n be a coordinate non commuting operator. In this respect, there can be no basis in Vn satisfying the Clifford algebra relation confusion with partial derivative, because the latter normally 1 acts on scalar fields, and in such a case partial derivative γμ γν (γμγν + γν γμ) = gμν , (49) and ∂ are one and the same thing. However, when acting ∙ ≡ 2 μ on a vector field, the derivative ∂μ is non commuting. Our where gμν is the metric of Vn. In curved space γμ and gμν μ μ operator ∂μ when acting on γμ or γ should be distinguished cannot be constant but necessarily depend on position x . from the ordinary — commuting — partial derivative, let be An arbitrary vector is a linear superposition [1] ν denoted γ ,μ , usually used in the literature on the Dirac μ equation in curved spacetime. The latter derivative is not a = a γμ , (50) used in the present paper, so there should be no confusion. μ where the components a are scalars from the geometric Using (55), eq.-(54) becomes point of view, whilst γμ are vectors. μ ν ν α μ ν μ ν Besides the basis γμ we can introduce the reciprocal ∂a =γ γν (∂μa +Γμαa ) γ γν Dμa =γ γ Dμaν (57) μ { } ≡ basis∗ γ satisfying { } where D is the covariant derivative of tensor analysis. 1 μ γμ γν (γμγν + γν γμ) = gμν , (51) Decomposing the Clifford product γμγν into its sym- ∙ ≡ 2 metric and antisymmetric part [1] μ ∗In Appendix A of the Hesteness book [1] the frame γ is called dual frame because the duality operation is used in constructing{ } it. γμγν = γμ γν + γμ γν , (58) ∙ ∧
42 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
where 4.1.2 Clifford algebra based geometric calculus and re- 1 γμ γν (γμγν + γν γμ) = gμν (59) solution of the ordering ambiguity for the product ∙ ≡ 2 of momentum operators is the inner product and Clifford algebra is a very useful tool for description of ge- ometry of curved space. Moreover, as shown in ref. [26] it 1 γμ γν (γμγν γν γμ) (60) provides a resolution of the long standing problem of the ∧ ≡ 2 − ordering ambiguity of quantum mechanics in curved space. the outer product, we can write eq.-(57) as Namely, eq.-(52) for the vector derivative suggests that the momentum operator is given by ∂a = gμν D a + γμ γν D a = μ ν μ ν μ ∧ (61) p = i ∂ = i γ ∂μ . (69) μ 1 μ ν − − = Dμa + γ γ (Dμaν Dν aμ) . 2 ∧ − One can consider three distinct models: (i) The non relativistic particle moving in ndimensional Without employing the expansion in terms of γμ we have simply curved space. Then, μ = 1, 2, . . . , n, and signature is (+ + + + ...); ∂a = ∂ a + ∂ a . (62) ∙ ∧ (ii) The relativistic particle in curved spacetime, described Acting twice on a vector by the operator ∂ we have∗ by the Schild action [37]. Then, μ = 0, 1, 2, . . . , n 1 and signature is (+ ...); − μ ν α μ ν α − − − ∂∂a = γ ∂μ(γ ∂ν )(a γα) = γ γ γαDμDν a = (iii) The Stueckelberg unconstrained particle [33, 34, 35, 1 29]. = γ D Dμaα + (γμ γν )γ [D , D ]aα = α μ 2 ∧ α μ ν (63) In all three cases the classical action has the form μ α μ ρ ρ α = γαDμD a + γ (Rμρa + Kμα Dρa ) + μ 1 μ ν I[X ] = dτ gμν (x)X˙ X˙ (70) 1 μ ν α ρ ρ α 2Λ + (γ γ γα)(Rμνρ a + Kμν Dρa ) . Z 2 ∧ ∧ and the corresponding Hamiltonian is We have used Λ Λ H = gμν (x)p p = p2 . (71) 2 μ ν 2 D D α α ρ ρD α (64) [ μ, ν ]a = Rμνρ a + Kμν ρa , If, upon quantization we take for the momentum operator p = i ∂ , then the ambiguity arises of how to write the where μ − μ ρ ρ ρ quantum Hamilton operator. The problem occurs because Kμν = Γ Γ (65) μν νμ μν μν μν − the expressions g pμpν , pμg pν and pμpν g are not α is torsion and Rμνρ the curvature tensor. Using eq.-(55) we equivalent. find But, if we rewrite H as [∂ , ∂ ]γ = R ν γ , (66) Λ α β μ αβμ ν H = p2 , (72) 2 from which we have μ where p = γ pμ is the momentum vector which upon quanti- ν ν zation becomes the momentum vector operator (69), we find Rαβμ = ([[∂α, ∂β]γμ) γ . (67) ∙ that there is no ambiguity in writing the square p2. When acting with H on a scalar wave function φ we obtain the Thus in general the commutator of derivatives ∂μ acting on a vector does not give zero, but is given by the curvature unambiguous expression tensor. Λ 2 Λ 2 μ ν Λ μ α1...αr Hφ = p φ = ( i) (γ ∂μ)(γ ∂ν )φ = DμD φ (73) In general, for an r-vector A = a γα1 γα2 . . . γαr we 2 2 − − 2 have in which there is no curvature term R. We expect that a term ∂∂ . . . ∂A = (γμ1 ∂ )(γμ2 ∂ ) ... (γμk ∂ ) with R will arise upon acting with H on a spinor field ψ. μ1 μ2 μk × α1...αr μ1 μ2 (a γα γα . . . γα ) = γ γ ... × 1 2 r (68) 4.2 C-space μk α1...αr . . . γ γα γα . . . γα Dμ Dμ ... Dμ a . 1 2 r 1 2 k Let us now consider C-space and review the procedure of ref. [20]. A basis in C-space is given by ∗We use (a b) c = (a b) c + a b c [1] and also (a b) c = = (b c) a (a ∧c)b. ∧ ∙ ∧ ∧ ∧ ∙ E = γ, γ , γ γ , γ γ γ ,... , (74) ∙ − ∙ A { μ μ ∧ ν μ ∧ ν ∧ ρ }
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where in an r-vector γ γ ... γ we take the indices In particular, for A = μ and E = γ we have μ1 ∧ μ2 ∧ ∧ μr A μ so that μ1 < μ2 < . . . < μr. An element of C-space is a Clifford number, called also Polyvector or Clifford aggregate ∂γμ ν ∂γμ αβ dγμ = dx + dx + ... = which we now write in the form ∂Xν ∂xαβ = Γ˜A E dxν + Γ˜A E dxαβ + ... = A μ μν νμ A [αβ]μ A (81) X = X EA = s γ + x γμ + x γμ γν + .... (75) ∧ = (Γ˜α γ + Γ˜[ρσ]γ γ + ...)dxν + νμ α νμ ρ ∧ σ A C-space is parametrized not only by 1-vector coordi- ρ [ρσ] αβ + (Γ˜ γρ + Γ˜ γρ γσ + ...)dx + .... nates xμ but also by the 2-vector coordinates xμν , 3-vector [αβ]μ [αβ]μ ∧ μνα coordinates x , etc., called also holographic coordinates, We see that the differential dγμ is in general a polyvector, since they describe the holographic projections of 1-loops, i. e., a Clifford aggregate. In eq.-(81) we have used 2-loops, 3-loops, etc., onto the coordinate planes. By p-loop we mean a closed p-brane; in particular, a 1-loop is closed ∂γμ ˜α ˜[ρσ] = Γνμγα + Γνμ γρ γσ + ..., (82) string. ∂xν ∧ In order to avoid using the powers of the Planck scale ∂γ μ = Γ˜ρ γ + Γ˜[ρσ] γ γ + .... (83) length parameter L in the expansion of the polyvector X we ∂xαβ [αβ]μ ρ [αβ]μ ρ ∧ σ use the dilatationally invariant units [15] in which L is set to 1. The dilation invariant physics was discussed from a Let us now consider a restricted space in which the ν αβ different perspective also in refs. [23, 21]. derivatives of γμ with respect to x and x do not contain In a flat C-space the basis vectors EA are constants. In a higher rank multivectors. Then eqs.-(82), (83) become curved C-space this is no longer true. Each EA is a function ∂γ μ = Γ˜α γ , (84) of the C-space coordinates ∂xν νμ α A μ μν X = s, x , x ,... (76) ∂γμ ρ { } = Γ˜ γρ . (85) ∂xαβ [αβ]μ which include scalar, vector, bivector, . . . , r-vector, . . . , co- Further we assume that: ordinates. C ˜α ˜C Now we define the connection Γ˜ in C-space accord- (i) The components Γνμ of the C-space connection ΓAB AB α ing to coincide with the connection Γνμ of an ordinary space; ˜C ∂AEB = ΓABEC , (77) ˜ρ (ii) The components Γ[αβ]μ of the C-space connection co- A ρ where ∂A ∂/∂X is the derivative in C-space. This incide with the curvature tensor Rαβμ of an ordinary definition ≡is analogous to the one in ordinary space. Let space. us therefore define the C-space curvature as Hence, eqs.-(84), (85) read
D D (78) ∂γμ ABC = ([∂A, ∂B]EC ) E , = Γα γ , (86) R ∗ ∂xν νμ α which is a straightforward generalization of the relation (67). ∂γ The “star” means the scalar product between two polyvectors μ ρ (87) αβ = Rαβμ γρ , A and B, defined as ∂x and the differential (81) becomes A B = AB S , (79) ∗ h i 1 dγ = Γρ dxα + R ρdxαβ γ . (88) where “S” means “the scalar part” of the geometric product μ αμ 2 αβμ ρ AB. The same relation was obtained by Pezzaglia [14] by In the following we shall explore the above relation for using a different method, namely by considering how poly- curvature and see how it is related to the curvature of the vectors change with position. The above relation demon- ordinary space. Before doing that we shall demonstrate that μν strates that a geodesic in C-space is not a geodesic in ordinary the derivative with respect to the bivector coordinate x is spacetime. Namely, in ordinary spacetime we obtain Papa- equal to the commutator of the derivatives with respect to μ petrou’s equation. This was previously pointed out by Pezza- the vector coordinates x . glia [14]. Returning now to eq.-(77), the differential of a -space C Although a C-space connection does not transform like basis vector is given by a C-space tensor, some of its components, i. e., those of eq.- (85), may have the transformation properties of a tensor in ∂EA B C B dEA = dX = Γ EC dX . (80) an ordinary space. ∂XB AB
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Under a general coordinate transformation in C-space The derivative of a basis vector with respect to the bivector coordinates xαβ is equal to the commutator of the A A A B X X0 = X0 (X ) (89) α → derivatives with respect to the vector coordinates x . The above relation (98) holds for the basis vectors γ . the connection transforms according to μ ∗ For an arbitrary polyvector C J K C 2 J C ∂X0 ∂X ∂X E ∂X0 ∂ X A α αβ ˜ ˜ (90) A = A EA = sγ + a γα + a γα γβ + ... (99) Γ0AB= E A B ΓJK + J A B . ∂X ∂X0 ∂X0 ∂X ∂X0 ∂X0 ∧ we will assume the validity of the following relation In particular, the components which contain the bivector index A = [αβ] transform as DAA = [D , D ]AA , (100) Dxμν μ ν ∂X ρ ∂XJ ∂XK ∂x ρ ∂2XJ ˜ ρ 0 ˜E 0 (91) Γ0[αβ]μ= E αβ μ ΓJK + J αβ μ . μν ∂X ∂σ0 ∂x0 ∂X ∂σ0 ∂x0 where D/Dx is the covariant derivative, defined in anal- Let us now consider a particular class of coordinate ogous way as in eqs. (57): transformations in C-space such that DAA ∂AA ˜A C (101) B = B + ΓBC A . ∂x ρ ∂xμν DX ∂X 0 (92) μν = 0 , α = 0 . ∂x ∂x0 From eq.-(100) we obtain Then the second term in eq.-(91) vanishes and the trans- Ds = [D , D ]s = K ρ∂ s , (102) formation becomes Dxμν μ ν μν ρ ∂X ρ ∂xρσ ∂xγ Daα ˜ ρ 0 ˜ (93) = [D , D ]aα = R αaρ + K ρD aα . (103) Γ0[αβ]μ = αβ μ Γ[ρσ]γ . μν μ ν μνρ μν ρ ∂x ∂σ0 ∂x0 Dx Now, for the bivector whose components are dxαβ we Using (101) we have that have Ds ∂s αβ αβ = (104) dσ0 γ0 γ0 = dx γα γβ . (94) Dxμν ∂xμν α ∧ β ∧ and also follows Taking into account that in our particular case (92) γα Daα ∂aα ∂aα transforms as a basis vector in an ordinary space = + Γ˜α aρ = + R αaρ , (105) Dxμν ∂xμν [μν]ρ ∂xμν μνρ ∂xμ (95) ˜α γα0 = α γμ , where, according to (ii), Γ[μν]ρ has been identified with ∂x0 curvature. So we obtain, after inserting (104), (105) into we find that (94) and (95) imply (102), (103) that: μ ν (a) The partial derivatives of the coefficients s and aα, αβ ∂x ∂x μν dσ0 = dx , (96) which are Clifford scalars , with respect to μν are ∂x α ∂x β † x 0 0 related to torsion: which means that ∂s ρ = Kμν ∂ρs , (106) ∂xμν 1 ∂xμ ∂xν ∂xν ∂xμ ∂x[μ ∂xν] ∂xμν = . (97) ∂σ αβ 2 ∂x α ∂x β −∂x α ∂x β ≡ ∂x α ∂x β ∂aα 0 0 0 0 0 0 0 ρD α (107) μν = Kμν ρa ; The transformation of the bivector coordinate xμν is thus ∂x determined by the transformation of the vector coordinates (b) Whilst the derivative of the basis vectors with respect μν xμ. This is so because the basis bivectors are the wedge to x are related to curvature: products of basis vectors γ . ∂γ μ α β (108) ˜ μν = Rμνα γβ . From (93) and (97) we see that Γ[ρσ]γ transforms like a ∂x 4th-rank tensor in an ordinary space. In other words, the dependence of coefficients s and aα Comparing eq.-(87) with the relation (66) we find on xμν indicates the presence of torsion. On the contrary, μν ∂γ when basis vectors γα depend on x this indicates that the μ = [∂ , ∂ ]γ . (98) ∂xαβ α β μ corresponding vector space has non vanishing curvature.
∂E0 In the geometric calculus based on Clifford algebra, the coefficients This can be derived from the relation dE = A dX B , where † ∗ A0 B 0 α αβ ∂X0 such as s, a , a ,. . . , are called scalars (although in tensor calculus D B ∂X B ∂X0 C they are called scalars, vectors and tensors, respectively), whilst the objects EA0 = ED and dX0 = dX . ∂X A ∂XC γα, γα γβ ,..., are called vectors, bivectors, etc. 0 ∧
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4.3 On the relation between the curvature of C-space the Universe [131], after a judicious choice of the algebraic and the curvature of an ordinary space coefficients is taken. One may notice also that having a vanishing cosmological constant in C-space, = Λ = 0 does Let us now consider the -space curvature defined in eq.- C not necessarily imply that one has a vanishingR cosmological (78). The indices , , can be of vector, bivector, etc., type. A B constant in ordinary spacetime. For example, in the very It is instructive to consider a particular example. special case of homogeneous symmetric spacetimes, like , , , A = [μν] B = [αβ] C = γ D = δ spheres and hyperboloids, where all the curvature tensors are proportional to suitable combinations of the metric tensor ∂ ∂ δ δ , γγ γ = [μν][αβ]γ . (109) times the scalar curvature, it is possible to envision that the ∂xμν ∂xαβ ∙ R net combination of the sum of all the powers of the curvature Using (87) we have tensors may cancel-out giving an overall zero value = 0. This possibility deserves investigation. R ∂ ∂ ∂ ρ ρ σ (110) Let us now show that for vanishing torsion the curvature μν αβ γγ = μν (Rαβγ γρ)=Rαβγ Rμνρ γσ ∂x ∂x ∂x is independent of the bivector coordinates xμν , as it was where we have taken taken in eq.-(111). Consider the basic relation
∂ ρ (114) R = 0 , (111) γμ γν = gμν . ∂xμν αβγ ∙ Differentiating with respect to xαβ we have which is true in the case of vanishing torsion (see also an ∂ ∂γμ ∂γν explanation that follows after the next paragraph). Inserting (γμ γν ) = γν + γμ = (110) into (109) we find ∂xαβ ∙ ∂xαβ ∙ ∙ ∂xαβ (115) = Rαβμν + Rαβνμ = 0 . δ = R ρR δ R ρR δ , (112) R[μν][αβ]γ μνγ αβρ − αβγ μνρ This implies that which is the product of two usual curvature tensors. We can ∂gμν = [∂α, ∂β]gμν = 0 . (116) proceed in analogous way to calculate the other components ∂σαβ D μ [μν] of ABC such as [αβγδ][ρσ] , [αβγδ][ρστκ] , etc. TheseR contain higher powersR of the curvatureR in an ordinary Hence the metric, in this particular case, is independent space. All this is true in our restricted C-space given by eqs.- of the holographic (bivector) coordinates. Since the curvature (84), (85) and the assumptions (i), (ii) bellow those equations. tensor — when torsion is zero — can be written in terms of By releasing those restrictions we would have arrived at an the metric tensor and its derivatives, we conclude that not even more involved situation which is beyond the scope of only the metric, but also the curvature is independent of μν the present paper. x . In general, when the metric has a dependence on the After performing the contractions of (112) and the corre- holographic coordinates one expects further corrections to sponding higher order relations we obtain the expansion of eq.-(112) that would include torsion. the form = R + α R2 + α R Rμν + .... (113) 5 On the quantization in C-spaces R 1 2 μν So we have shown that the C-space curvature can be ex- 5.1 The momentum constraint in C-space pressed as the sum of the products of the ordinary spacetime A detailed discussion of the physical properties of all the curvature. This bears a resemblance to the string effective components of the polymomentum P in four dimensions and action in curved spacetimes given by sums of powers of the emergence of the physical mass in Minkowski spacetime the curvature tensors based on the quantization of non-linear has been provided in the book [15]. The polymomentum in sigma models [118]. D = 4, canonically conjugate to the position polyvector If one sets aside the algebraic convergence problems μ μν μ when working with Clifford algebras in infinite dimensions, X = σ + x γμ + γ γμ γν + ξ γ5γμ + sγ5 (117) one can consider the possibility of studying Quantum Gravity ∧ in a very large number of dimensions which has been revi- can be written as: sited recently [83] in connection to a perturbative renorm- μ μν μ P = μ + p γμ + S γμ γν + π γ5γμ + mγ5 , (118) alizable quantum theory of gravity in infinite dimensions. ∧ Another interesting possibility is that an infinite series ex- where besides the vector components pμ we have the scalar pansion of the powers of the scalar curvature could yield the component μ, the 2-vector components Sμν , that are con- recently proposed modified Lagrangians R + 1/R of gravity nected to the spin as shown by [14]; the pseudovector com- to accommodate the cosmological accelerated expansion of ponents πμ and the pseudoscalar component m.
46 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
The most salient feature of the polyparticle dynamics in condition by replacing C-spaces [15] is that one can start with a constrained action ∂ ∂ ∂ ∂ in C-space and arrive, nevertheless, at an unconstrained P i = i , , ,... , (121) A → − ∂XA − ∂σ ∂xμ ∂xμν Stuckelberg action in Minkowski space (a subspace of C- space) in which p pμ is a constant of motion. The true μ ∂2 ∂2 ∂2 constraint in C-space is: + + + ... + M 2 Φ = 0, (122) ∂σ2 ∂xμ∂x ∂xμν ∂x μ μν A 2 μ μν μ 2 2 PAP = μ +pμp 2S Sμν +πμπ m = M , (119) where we have set for convenience purposes − − L = ~ = c = 1 and the C-space scalar field Φ(σ, xμ, xμν ,...) is a poly- where M is a fixed constant, the mass in C-space. The vector-valued scalar function of all the C-space variables. μν pseudoscalar component m is a variable, like μ, pμ, S , and This is the Klein-Gordon equation associated with a free μ π , which altogether are constrained according to eq.-(119). scalar polyparticle in C-space. It becomes the physical mass in Minkowski spacetime in the A wave equation for a generalized C-space harmonic special case when other extra components vanish, i. e., when oscillator requires to introduce the potential of the form μν μ μ = 0, S = 0 and π = 0. This justifies using the notation V = κX2 that admits straightforward solutions in terms of m for mass. This is basically the distinction between the mass Gaussians and Hermite polynomials similar to the ordinary μ in Minkowski space which is a constant of motion pμp and point-particle oscillator. There are now collective excitations the fixed mass M in C-space. The variable m is canonically of the Clifford-oscillator in terms of the number of Clifford- conjugate to s which acquires the role of the Stuckelberg bits and which represent the quanta of areas, volumes, hyper- evolution parameter s that allowed ref. [29, 15] to propose a volumes, . . . , associated with the p-loops oscillations in natural solution of the problem of time in quantum gravity. Planck scale units. The logarithm of the degeneracy of the The polyparticle dynamics in C-space is a generalization of first collective state of the C-space oscillator, as a function the relativistic Regge top construction which has recently of the number of bits, bears the same functional form as the been studied in de Sitter spaces by [135]. Bekenstein-Hawking black hole entropy, with the upshot that A derivation of a charge, mass, and spin relationship of a one recovers, in a natural way, the logarithmic corrections to polyparticle can be obtained from the above polymomentum the black-hole entropy as well, if one identifies the number constraint in C-space if one relates the norm of the axial- of Clifford-bits with the number of area-quanta of the black μ momentum component π of the polymomentum P to the hole horizon. For further details about this derivation and charge [80]. It agrees exactly with the recent charge-mass- the emergence of the Schwarzschild horizon radius relation, spin relationship obtained by [44] based on the Kerr- the Hawking temperature, the maximal Planck temperature Newman black hole metric solutions of the Einstein-Maxwell condition, etc., we refer to [21]. Perhaps the most important equations. The naked singularity Kerr-Newman solutions consequence of this latter view of black hole entropy is the have been interpreted by [45] as Dirac particles. Further possibility that there is a ground state of quantum spacetime, investigation is needed to understand better these relation- resulting from of a Bose-Einstein condensate of the C-space ships, in particular, the deep reasons behind the charge as- harmonic oscillator. μ signment to the norm of the axial-vector π component of the A C-space version of the Dirac Equation, representing polymomentum which suggests that mass has a gravitational, the dynamics of spinning-polyparticles (theories of extended- electromagnetic and rotational aspects to it. In a Kaluza- spin, extended charges) is obtained via the square-root pro- Klein reduction from D = 5 to D = 4 it is well known that cedure of the Klein-Gordon equation: the electric charge is related to the p5 component of the momentum. Hence, charge bears a connection to an internal ∂ ∂ ∂ i +γμ +γμ γν + ... Ψ = MΨ, (123) momentum. − ∂σ ∂x ∧ ∂x μ μν where Ψ(σ, xμ, xμν ,...) is a polyvector-valued function, a 5.2 C-space Klein-Gordon and Dirac wave equations A Clifford-number, Ψ = Ψ EA of all the C-space variables. The ordinary Klein-Gordon equation can be easily obtained For simplicity we consider here a flat C-space in which the by implementing the on-shell constraint p2 m2 = 0 as an metric G = E† E = η is diagonal, η being the − AB A ∗ B AB AB operator constraint on the physical states after replacing pμ C-space analog of Minkowski tensor. In curved C-space the for i∂/∂xμ (we use units in which ~ = 1, c = 1): equation (123) should be properly generalized. This goes − beyond the scope of the present paper. ∂2 Ordinary spinors are nothing but elements of the left/right + m2 φ = 0. (120) ∂xμ∂x ideals of a Clifford algebra. So they are automatically con- μ tained in the polyvector valued wave function Ψ. The ordi- The C-space generalization follows from the P 2 M 2=0 nary Dirac equation can be obtained when Ψ is independent −
C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 47 Volume 1 PROGRESS IN PHYSICS April, 2005
of the extra variables associated with a polyvector-valued But on the other hand, the introduction of a bivector into an coordinates X (i. e., of xμν , xμνρ, . . . ). For details see [15]. equation implies a selection of a preferred orientation in Thus far we have written ordinary wave equations in spacetime; i. e. the choice of the spin quantization axis in the C-space, that is, we considered the wave equations for a original Dirac-Lanczos quaternionic equation. How is such “point particle” in C-space. From the perspective of the 4- preferred orientation (spin quantization axis) determined? dimensional Minkowski spacetime the latter “point particle” Is there some dynamical symmetry which determines the has, of course, a much richer structure then a mere point: it preferred orientation (spin quantization axis)? is there an is an extended object, modeled by coordinates xμ, xμν ,... action which encodes a hidden dynamical principle that se- But such modeling does not embrace all the details of an lects dynamically a preferred spacetime orientation (spin extended object. In order to provide a description with more quantization axis)? details, one can considere not the “point particles” in C- Many subtleties of the Dirac-Hesteness equation and its space, but branes in C-space. They are described by the relation to the ordinary Dirac equation and the Seiberg- embeddings X = X(Σ), that is XM = XM (ΣA), considered Witten equation are investigated from the rigorous mathe- in sec. 3.2. Quantization of such branes can employ wave matical point of view in refs. [93]. The approach in refs. [16, functional equation, or other methods, including the second 15, 17, 8], reviewed here, is different. We start from the usual quantization formalism. For a more detailed study detailed formulation of quantum theory and extend it to C-space. We study of the second quantization of extended objects using retain the imaginary unit i. Next step is to give a geometric the tools of Clifford algebra see [15]. interpretation to i. Instead of trying to find a geometric origin Without emplying Clifford algebra a lot of illuminating of i in spacetime we adopt the interpretation proposed in [15] work has been done in relation to description of branes in according to which the i is the bivector of the 2-dimensional terms of p-loop coordinates [132]. A bosonic/fermionic p- phase space (whose direct product with the n-dimensional brane wave-functional equation was presented in [12], gener- configuration space gives the 2n-dimensional phase space)∗. alizing the closed-string (loop) results in [13] and the the This appears to be a natural assumption due to the fact that quantum bosonic p-brane propagator, in the quenched- complex valued quantum mechanical wave functions involve μ reduced minisuperspace approximation, was attained by [18]. momenta pμ and coordinates x (e. g., a plane wave is given μ In the latter work branes are described in terms of the by exp[ipμx ], and arbitrary wave packet is a superposition collective coordinates which are just the highest grade com- of plane waves). ponents in the expansion of a poplyvector X given in eq.-(2). This work thus paved the way for the next logical step, that is, to consider other multivector components of X in a unified 6 Maximal-acceleration Relativity in phase-spaces description of all branes. Notice that the approach based on eqs.-(122), (123) is In this section we shall discuss the maximal acceleration different from that by Hestenes [1] who proposed an equation Relativity principle [68] based on Finsler geometry which which is known as the Dirac-Hestenes equation. Dirac’s does not destroy, nor deform, Lorentz invariance. Our dis- equation using quaternions (related to Clifford algebras) was cussion differs from the pseudo-complex Lorentz group de- first derived by Lanczos [91]. Later on the Dirac-Lanczos scription by Schuller [61] related to the effects of maximal equation was rediscovered by many people, in particular by acceleration in Born-Infeld models that also maintains Lo- Hestenes and Gursey [92] in what became known as the rentz invariance, in contrast to the approaches of Double Dirac-Hestenes equation. The former Dirac-Lanczos equa- Special Relativity (DSR). In addition one does not need to tion is Lorentz covariant despite the fact that it singles out modify the energy-momentum addition (conservation) laws an arbitrary but unique direction in ordinary space: the spin in the scattering of particles which break translational invari- quantization axis. Lanczos, without knowing, had anticipated ance. For a discussions on the open problems of Double Spe- the existence of isospin as well. The Dirac-Hestenes equation cial Relativity theories based on kappa-deformed Poincare´ ∂Ψe21 = mΨe0 is covariant under a change of frame [133], symmetries [63] and motivated by the anomalous Lorentz- 1 1 [93]. eμ0 = UeμU − and Ψ0 = ΨU − with U an element of violating dispersion relations in the ultra high energy cosmic the Spin+(1, 3) yielding ∂Ψ0e210 = mΨ0e0. As Lanczos had rays [71, 72, 73], we refer to [70]. anticipated, in a new frame of reference, the spin quantization Related to the minimal Planck scale, an upper limit on the axis is also rotated appropriately, thus there is no breakdown maximal acceleration principle in Nature was proposed by of covariance by introducing bivectors in the Dirac-Hestenes long ago Cainello [52]. This idea is a direct consequence of a equation. suggestion made years earlier by Max Born on a Dual Relati- However, subtleties still remain. In the Dirac-Hestenes vity principle operating in phase spaces [49], [74] wherethere equation instead of the imaginary unit i there occurs the Yet another interpretation of the imaginary unit i present in the bivector γ γ 2. Its square is 1 and commutes with all the ∗ 1 − − Heisenberg uncertainty relations has been undertaken by Finkelstein and elements of the Dirac algebra which is just a desired property. collaborators [96].
48 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
is an upper bound on the four-force (maximal string tension i = ep eq, a phase-space area element. Such imaginary unit or tidal forces in the string case) acting on a particle as well as allows∧ us to express vectors in a C-phase space in the form: an upper bound in the particle velocity. One can combine the maximum speed of light with a minimum Planck scale into a Q = qeq + peq , maximal proper-acceleration a = c2/L within the framework Q eq = q + pep eq = q + ip = z , (126) of Finsler geometry [56]. For a recent status of the geometries ∙ ∙ e Q = q + pe e = q ip = z∗ , behind maximal-acceleration see [73]; its relation to the q ∙ q ∙ p − Double Special Relativity programs was studied by [55] and which reminds us of the creation/annihilation operators used the possibility that Moyal deformations of Poincare´ algebras in the harmonic oscillator. could be related to the kappa-deformed Poincare´ algebras We shall now review the steps in [127] to reproduce the was raised in [68]. A thorough study of Finsler geometry sub-maximally accelerated particle action [53]. The phase- and Clifford algebras has been undertaken by Vacaru [81] space analog of the spacetime action is: where Clifford/spinor structures were defined with respect to Nonlinear connections associated with certain nonholonomic dQdQ = (dq)2 +(dp)2 S = m (dq)2 +(dp)2 . (127) modifications of Riemann-Cartan gravity. ⇒ Other several new physical implications of the maximal Zp acceleration principle in Nature, like neutrino oscillations Introducing the appropriate length/mass scale parameters and other phenomena, have been studied by [54], [67], [42]. in order to have consistent units yields: Recently, the variations of the fine structure constant α [64], with the cosmological accelerated expansion of the Universe, L 2 S = m (dq)2 + (dp)2 , (128) was recast as a renormalization group-like equation govern- s m ing the cosmological red shift (Universe scale) variations of Z α based on this maximal acceleration principle in Nature where we have introduced the Planck scale L and have [68]. The fine structure constant was smaller in the past. chosen the natural units ~ = c = 1. A detailed physical dis- Pushing the cutoff scale to the minimum Planck scale led cussion of the dilational invariant system of units ~ = c = to the intriguing result that the fine structure constant could = G = 4π0 = 1 was presented in ref. [15]. G is the Newton have been extremely small (zero) in the early Universe and constant and 0 is the permittivity of the vacuum. that all matter in the Universe could have emerged via the Extending this two-dim result to a 2n-dim phase space
Unruh-Rindler-Hawking effect (creation of radiation/matter) result requires to have for Clifford basis the elements epμ , due to the acceleration w. r. t the vacuum frame of reference. eqμ , where μ = 1, 2, 3, . . . n. The action in the 2n-dim phase For reviews on the alleged variations of the fundamental space is: constants in Nature see [65] and for more astonishing vari- ations of αdriven by quintessence see [66]. 2 μ L μ S = m (dq dqμ) + (dp dpμ) = s m Z 6.1 Clifford algebras in phase space (129) 2 L μ We shall employ the procedure described in [15] to construct = m dτ 1 + (dp /dτ)(dpμ/dτ) , s m the Phase Space Clifford algebra that allowed [127] to repro- Z duce the sub-maximally accelerated particle action of [53]. where we have factored-out of the square-root the infinite- 2 μ For simplicity we will focus on a two-dim phase space. simal proper-time displacement (dτ) = dq dqμ. Let ep, eq be the Clifford-algebra basis elements in a two-dim One can recognize the action (129), up to a numerical phase space obeying the following relations [15]: factor of m/a, where a is the proper acceleration, as the same action for a sub-maximally accelerated particle given 1 e e (e e + e e ) = 0 (124) by Nesterenko [53] by rewriting (dpμ/dτ) = m(d2xμ/dτ 2): p ∙ q ≡ 2 q p p q
2 2 μ 2 2 2 and epep = eqeq = 1. S = m dτ 1 + L (d x /dτ )(d xμ/dτ ) . (130) The Clifford product of ep, eq is by definition the sum of Z q the scalar and the wedge product: Postulating that the maximal proper-acceleration is given in terms of the speed of light and the minimal Planck scale e e = e e + e e = 0 + e e = i , (125) 2 p q p ∙ q p ∧ q p ∧ q by a = c /L = 1/L, the action above gives the Nesterenko 2 action, up to a numerical m/a factor: such that i = epeqepeq = 1. Hence, the imaginary unit i, i2 = 1 admits a very natural− interpretation in terms of − 2 2 μ 2 2 2 Clifford algebras, i. e., it is represented by the wedge product S = m dτ 1 + a− (d x /dτ )(d xμ/dτ ) . (131) Z q C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 49 Volume 1 PROGRESS IN PHYSICS April, 2005
The proper-acceleration is orthogonal to the proper- 6.2 Invariance under the U(1, 3) Group velocity and this can be easily verified by differentiating In this section we will review in detail the principle of the time-like proper-velocity squared: Maximal-acceleration Relativity [68] from the perspective μ dx dxμ of 8D Phase Spaces and the U (1, 3) Group. The U (1, 3) = V 2 = = V μV = 1 > 0 dτ dτ μ ⇒ = SU (1, 3) U(1) Group transformations, which leave in- (132) ⊗ dV μ d2xμ variant the phase-space intervals under rotations, velocity V = V = 0 , ⇒ dτ μ dτ 2 μ and acceleration boosts, were found by Low [74] and can be simplified drastically when the velocity/acceleration boosts which implies that the proper-acceleration is space-like: are taken to lie in the z-direction, leaving the transverse direc- 2 μ 2 tions x, y, px, py intact; i. e., the U (1, 1) = SU (1, 1) U (1) 2 d x d xμ g (τ) = > 0 subgroup transformations that leave invariant the⊗ phase- − dτ 2 dτ 2 ⇒ (133) space interval are given by (in units of ~ = c = 1) g2 S = m dτ 1 = m dω , 2 (dE)2 (dP )2 ⇒ r − a 2 2 2 Z Z (dσ) = (dT ) (dX) + −2 = where the analog of the Lorentz time-dilation factor for a − b sub-maximally accelerated particle is given by (dE/dτ)2 (dP/dτ)2 = (dτ)2 1 + − = (136) b2 g2(τ) (134) 2 2 dω = dτ 1 2 . m g (τ) r − a = (dτ)2 1 , − m2 A2 Therefore the dynamics of a sub-maximally accelerated P max particle can be reinterpreted as that of a particle moving in where we have factored out the proper time infinitesimal the spacetime tangent bundle whose Finsler-like metric is (dτ)2 = dT 2 dX2 in eq.-(136) and the maximal proper- force is set− to be b m A . m is the Planck mass g2(τ) P max P (dω)2 =g (xμ, dxμ)dxμdxν =(dτ)2 1 . (135) 1/L so that b = (1/L≡ )2, may also be interpreted as the μν − a2 P P maximal string tension when LP is the Planck scale. The invariant time now is no longer the standard proper- The quantity g(τ) is the proper four-acceleration of a time τ but is given by the quantity ω(τ). The deep connection particle of mass m in the z-direction which we take to be between the physics of maximal acceleration and Finsler X. Notice that the invariant interval (dσ)2 in eq.-(136) is geometry has been analyzed by [56]. This sort of actions not strictly the same as the interval (dω)2 of the Nesterenko involving second derivatives have also been studied in the action eq.-(131), which was invariant under a pseudo- construction of actions associated with rigid particles complexification of the Lorentz group [61]. Only when 2 (strings) [57], [58], [59], [60] among others. m = mP , the two intervals agree. The interval (dσ) de- The action is real-valued if, and only if, g2 < a2 in the scribed by Low [74] is U (1, 3)-invariant for the most general same fashion that the action in Minkowski spacetime is real- transformations in the 8D phase-space. These transforma- valued if, and only if, v2 < c2. This is the physical reason tions are rather elaborate, so we refer to the references [74] why there is an upper bound in the proper-acceleration. In for details. The analog of the Lorentz relativistic factor in eq.- the special case of uniformly-accelerated motion g(τ) = g0 = (136) involves the ratios of two proper forces. One variable = constant, the trajectory of the particle in Minkowski space- force is given by ma and the maximal proper force sustained time is a hyperbola. by an elementary particle of mass mP (a Planckton) is 2 Most recently, an Extended Relativity Theory in Born- assumed to be Fmax = mP lanckc /LP . When m = mP , the Clifford-Phase spaces with an upper and lower length scales ratio-squared of the forces appearing in the relativistic factor 2 2 (infrared/ultraviolet cutoff ) has been constructed [138]. The of eq.-(136) becomes then g /Amax, and the phase space invariance symmetry associated with an 8D Phase Space interval (136) coincides with the geometric interval of (131). leads naturally to the real Clifford algebra Cl(2, 6,R) and The transformations laws of the coordinates in that leave complexified Clifford ClC (4) algebra related to Twistors. invariant the interval (136) are [74]: The consequences of Mach’s principle of inertia within the ξvX ξaP sinh ξ context of Born’s Dual Phase Space Relativity Principle T 0 = T cosh ξ + + , (137) c2 b2 ξ were also studied in [138] and they were compatible with the Eddington-Dirac large numbers coincidence and with sinh ξ E0 = E cosh ξ + ( ξ X + ξ P ) , (138) the observed values of the anomalous Galileo-Pioneer ac- − a v ξ celeration. The modified Newtonian dynamics due to the upper/lower scales and modified Schwarzschild dynamics ξaE sinh ξ X0 = X cosh ξ + ξ T , (139) due the maximal acceleration were also provided. v − b2 ξ
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(136) but does not leave invariant the proper time interval ξvE sinh ξ P 0 = P cosh ξ + + ξaT . (140) 2 2 2. Only the combination: c2 ξ (dτ) = dT dX − The is velocity-boost rapidity parameter and the m2g2 ξv ξa (dσ)2 = (dτ)2 1 (148) is the force/acceleration-boost rapidity parameter of the − m2 A2 P max primed-reference frame. They are defined respectively (in is truly left invariant under pure acceleration-boosts (144)– the special case when m = mP ): (146). One can verify as well that these transformations ξ v satisfy Born’s duality symmetry principle: tanh v = , c c (T,X) (E,P ) , (E,P ) ( T, X) (149) → → − − ξa ma tanh = . (141) and b 1 . The latter Born duality transformation is nothing b mP Amax b but a→ manifestation of the large/small tension duality prin- The effective boost parameter ξ of the U (1, 1) subgroup ciple reminiscent of the T -duality symmetry in string theory; transformations appearing in eqs.-(137)–(140) is defined in i. e. namely, a small/large radius duality, a winding modes/ terms of the velocity and acceleration boosts parameters ξv, Kaluza-Klein modes duality symmetry in string compactifi- ξa respectively as: cations and the Ultraviolet/Infrared entanglement in Non- commutative Field Theories. Hence, Born’s duality prin- ξ2 ξ2 v a (142) ciple in exchanging coordinates for momenta could be the ξ 2 + 2 . ≡ r c b underlying physical reason behind T -duality in string theory. Our definition of the rapidity parameters are different The composition of two successive pure acceleration- than those in [74]. boosts is another pure acceleration-boost with acceleration Straightforward algebra allows us to verify that these rapidity given by ξ00 = ξ + ξ0. The addition of proper four- transformations leave the interval of eq.-(136) in classical forces (accelerations) follows the usual relativistic compo- phase space invariant. They are are fully consistent with sition rule: Born’s duality Relativity symmetry principle [49] (Q, P ) → tanh ξ + tanh ξ0 (P, Q). By inspection we can see that under Born dual- tanh ξ00 = tanh(ξ + ξ0) = 1 + tanh ξ tanh ξ ⇒ →ity, the− transformations in eqs.-(137)–(140) are rotated into 0 (150) each other, up to numerical b factors in order to match ma ma + ma0 00 = mP A mP A , units. When on sets ξa = 0 in (137)–(140) one recovers m2aa ⇒ mP A 0 1 + m2 A2 automatically the standard Lorentz transformations for the P X,T and E,P variables separately, leaving invariant the and in this fashion the upper limiting proper acceleration is intervals dT 2 dX2 = (dτ)2 and (dE2 dP 2)/b2 separately. never surpassed like it happens with the ordinary Special When one− sets we obtain the− transformations rules ξv = 0 Relativistic addition of velocities. of the events in Phase space, from one reference-frame into The group properties of the full combination of velocity another uniformly-accelerated frame of reference, a = const, and acceleration boosts (137)–(140) requires much more whose acceleration-rapidity parameter is in this particular algebra [68]. A careful study reveals that the composition case: rule of two succesive full transformations is given by ξ = ξa ma 00 ξ , tanh ξ = . (143) = ξ + ξ0 and the transformation laws are preserved if, and ≡ b mP Amax only if, the ξ ; ξ0 ; ξ00 . . . parameters obeyed the suitable The transformations for pure acceleration-boosts in are: relations: ξa ξa0 ξa00 ξa00 P = = = , (151) T 0 = T cosh ξ + sinh ξ , (144) ξ ξ0 ξ00 ξ + ξ0 b ξ ξ ξ ξ v = v0 = v00 = v00 . (152) E0 = E cosh ξ bX sinh ξ , (145) ξ ξ ξ ξ + ξ − 0 00 0 E Finally we arrive at the composition law for the effective, X0 = X cosh ξ sinh ξ , (146) − b velocity and acceleration boosts parameters ξ00; ξv00; ξa00 re- spectively: ξ00 = ξ + ξ0 , (153) P 0 = P cosh ξ + bT sinh ξ . (147) v v v ξ = ξ + ξ , (154) It is straightforward to verify that the transformations a00 a a0 (144)–(146) leave invariant the fully phase space interval ξ00 = ξ + ξ0 . (155)
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2 2 The relations (151, 152, 153, 154, 155) are required in ΔP 0ΔE0 = 0 = ΔP ΔE(cosh ξ sinh ξ) = ∞× − order to prove the group composition law of the transfor- 1 (164) = ΔP ΔE = 2 . mations of (137)–(140) and, consequently, in order to have LP a truly Maximal-Acceleration Phase Space Relativity theory resulting from a phase-space change of coordinates in the It is important to emphasize that the invariance property cotangent bundle of spacetime. of the minimal Planck-scale Areas (maximal Tension) is not an exclusive property of infinite acceleration boosts ξ = , but, as a result of the identity cosh2 ξ sinh2 ξ = 1,∞ for 6.3 Planck-Scale Areas are invariant under acceleration − boosts all values of ξ, the minimal Planck-scale Areas are always invariant under any acceleration-boosts transformations. Having displayed explicitly the Group transformations rules Meaning physically, in units of ~ = c = 1, that the Maximal 1 of the coordinates in Phase space we will show why infinite Tension (or maximal Force) b = 2 is a true physical invar- LP acceleration-boosts (which is not the same as infinite proper iant universal quantity. Also we notice that the Phase-space acceleration) preserve Planck-Scale Areas [68] as a result of areas, or cells, in units of , are also invariant! The pure- 2 ~ the fact that b = (1/LP ) equals the maximal invariant force, acceleration boosts transformations are “symplectic”. It can or string tension, if the units of ~ = c = 1 are used. be shown also that areas greater (smaller) than the Planck- At Planck-scale LP intervals/increments in one reference area remain greater (smaller) than the invariant Planck-area frame we have by definition (in units of ~ = c = 1): ΔX = under acceleration-boosts transformations. 1 1 = ΔT = LP and ΔE = ΔP = L where b L2 is the max- P ≡ P The infinite acceleration-boosts are closely related tothe imal tension. From eqs.-(137)–(140) we get for the trans- infinite red-shift effects when light signals barely escape formation rules of the finite intervals ΔX, ΔT , ΔE, ΔP , Black hole Horizons reaching an asymptotic observer with an from one reference frame into another frame, in the infinite infinite red shift factor. The important fact is that the Planck- acceleration-boost limit ξ , → ∞ scale Areas are truly maintained invariant under acceleration- boosts. This could reveal very important information about ΔT 0 = L (cosh ξ + sinh ξ) , (156) P → ∞ Black-holes Entropy and Holography. The logarithmic cor- 1 rections to the Black-Hole Area-Entropy relation were ob- ΔE0 = (cosh ξ sinh ξ) 0 (157) LP − → tained directly from Clifford-algebraic methods in C-spaces by a simple use of L’Hopital’sˆ rule or by noticing that both [21], in addition to the derivation of the maximal Planck cosh ξ; sinh ξ functions approach infinity at the same rate temperature condition and the Schwarzschild radius in terms of the Thermodynamics of a gas of p-loop-oscillators quanta ΔX0 = LP (cosh ξ sinh ξ) 0 , (158) represented by area-bits, volume-bits, . . . hyper-volume-bits − → in Planck scale units. Minimal loop-areas, in Planck units, is 1 ΔP 0 = (cosh ξ + sinh ξ) , (159) also one of the most important consequences found in Loop LP → ∞ Quantum Gravity long ago [111]. where the discrete displacements of two events in Phase Spa- 1 ce are defined: ΔX = X2 X1 = LP , ΔE = E2 E1 = , − − LP ΔT = T T = L and ΔP = P P = 1 . 7 Some further important physical applications related 2 1 P 2 1 LP Due to− the identity: − to the C-space physics
(cosh ξ+ sinh ξ)(cosh ξ sinh ξ)= cosh2 ξ sinh2 ξ=1 (160) − − 7.1 Relativity of signature one can see from eqs.-(156)–(159) that the Planck-scale In previous sections we have seen how Clifford algebra can Areas are truly invariant under infinite acceleration-boosts be used in the formulation of the point particle classical ξ = : ∞ and quantum theory. The metric of spacetime was assumed, 2 2 as usually, to have the Minkowski signature, and we have ΔX0ΔP 0 = 0 = ΔXΔP (cosh ξ sinh ξ) = used the choice . There were arguments in the ×∞ − L (161) (+ ) = ΔXΔP = P = 1 , literature of why the spacetime− −− signature is of the Minkowski L P type [113, 43]. But there are also studies in which signature 2 2 ΔT 0ΔE0 = 0 = ΔT ΔE(cosh ξ sinh ξ) = changes are admitted [112]. It has been found out [16, 15, 30] ∞× − LP (162) that within Clifford algebra the signature of the underlying = ΔT ΔE = = 1 , space is a matter of choice of basis vectors amongst available LP 2 2 Clifford numbers. We are now going to review those impor- ΔX0ΔT 0 = 0 = ΔXΔT (cosh ξ sinh ξ) = tant topics. ×∞ − (163) 2 = ΔXΔT = (LP ) , Suppose we have a 4-dimensional space V4 with signature
52 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
(+ + ++). Let eμ, μ = 0, 1, 2, 3, be basis vectors satisfying As an alternative to (167) we can choose
1 e0e3 γ˜0 , eμ eν (eμeν + eν eμ) = δμν , (165) ≡ (172) ∙ ≡ 2 e γ˜ , i ≡ i where δμν is the Euclidean signature of V4. The vectors eμ from which we have can be used as generators of Clifford algebra 4 over V4 with a generic Clifford number (also called polyvectorC or Clifford 1 (˜γμγ˜ν +γ ˜ν γ˜μ) =η ˜μν (173) aggregate) expanded in term of eJ =(1,eμ,eμν ,eμνα,eμναβ), 2 , μ < ν < α < β with η˜ = diag( 1, 1, 1, 1). Obviously γ˜ are basis vectors μν − μ J μ μν of a pseudo-Euclidean space V1,3 and they generate the A = a eJ = a + a eμ + a eμeν + (166) Clifford algebra over V1,3 which is yet another representation μνα μναβ + a eμeν eα + a eμeν eαeβ . of the same set of objects (i. e., polyvectors).e The spaces V4, V1,3 and V1,3 are differente slices through C-space, and they Let us consider the set of four Clifford numbers (e0, eie0), span different subsets of polyvectors. In a similar way we can , and denote them as i = 1, 2, 3 obtain spacese with signatures (+ ++), (+ + +), (+ + + ), ( + ), ( + ), ( − +) and corresponding− e0 γ0 , − − −− − − − − − − ≡ (167) higher dimensional analogs. But we cannot obtain signatures eie0 γi . of the type (+ + ), (+ + ), etc. In order to obtain ≡ such signatures we−− proceed as− follows.− The Clifford numbers γμ, μ = 0, 1, 2, 3, satisfy 4-space. First we observe that the bivector Iˉ4-space. e e satisfies Iˉ2 = 1, commutes with e , e and anticom- 1 3 4 − 1 2 (γ γ + γ γ ) = η , (168) mutes with e3, e4. So we obtain that the set of Clifford 2 μ ν ν μ μν numbers γμ = (e1I,ˉ e2I,ˉ e3, e3) satisfies where ημν = diag(1, 1, 1, 1) is the Minkowski tensor. − − − γμ γν =η ˉμν , (174) We see that the γμ behave as basis vectors in a 4-dimensional ∙ space V1,3 with signature (+ ). We can form a Clifford where ηˉ = diag( 1, 1, 1, 1). − −− − − aggregate 8-space. Let eA be basis vectors of 8-dimensional μ α = α γμ , (169) vector space with signature (+ + + + + + + +). Let us decompose which has the properties of a vector in V1,3. From the point of view of the space V4 the same object α is a linear eA = (eμ, eμˉ) , μ = 0, 1, 2, 3 , combination of a vector and bivector: (175) μˉ = 0ˉ, 1ˉ, 2ˉ, 3ˉ . 0 i α = α e0 + α eie0 . (170) The inner product of two basis vectors
We may use γμ as generators of the Clifford al- e e = δ , (176) A ∙ B AB gebra 1,3 defined over the pseudo-Euclidean space V1,3. The basis elementsC of are γ = (1, γ , γ , γ , γ ), then splits into the following set of equations: C1,3 J μ μν μνα μναβ with μ < ν < α < β. A generic Clifford aggregate in 1,3 is eμ eν = δμν , given by C ∙ eμˉ eνˉ = δμˉνˉ , (177) J μ μν ∙ B = b γJ = b + b γμ + b γμγν + (171) eμ eνˉ = 0 . μνα μναβ ∙ + b γμγν γα + b γμγν γαγβ . The number Iˉ= e0ˉe1ˉe2ˉe3ˉ has the properties With suitable choice of the coefficients bJ = (b, bμ, bμν , Iˉ2 = 1 , bμνα, bμναβ) we have that B of eq.-(171) is equal to A of eq.-(166). Thus the same number A can be described either Ieˉ μ = eμI,ˉ (178) with eμ which generate 4, or with γμ which generate 1,3. C C Ieˉ μˉ = eμˉI.ˉ The expansions (171) and (166) exhaust all possible numbers − of the Clifford algebras 1,3 and 4. Those expansions are The set of numbers just two different representationsC ofC the same set of Clifford γμ = eμ , numbers (also being called polyvectors or Clifford ag- (179) gregates). γμˉ = eμˉIˉ
C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 53 Volume 1 PROGRESS IN PHYSICS April, 2005
satisfies 7.2 Clifford space and the conformal group γμ γν = δμν , ∙ 7.2.1 Line element in C-space of Minkowski spacetime γμˉ γνˉ = δμν , (180) ∙ − In 4-dimensional spacetime a polyvector and its square (1) γ γ = 0 . μ ∙ μˉ can be written as The numbers (γ , γ ) thus form a set of basis vectors of μ μˉ μ 1 μν μ a vector space V with signature (+ + + + ). dX = dσ+dx γμ + dx γμ γν +dx˜ Iγμ +dσI˜ , (184) 4,4 − − −− 2 ∧ 10-space. Let eA = (eμ, eμˉ), μ = 1, 2, 3, 4, 5; μˉ = ˉ ˉ ˉ ˉ ˉ = 1, 2, 3, 4, 5 be basis vectors of a 10-dimensional Euclid- 2 2 μ 1 μν μ 2 ˉ dX = dσ +dx dxμ+ dx dxμν dx˜ dx˜μ dσ˜ . (185) ean space V10 with signature (+ + + ...). We introduce I = | | 2 − − = e1ˉe2ˉe3ˉe4ˉe5ˉ which satisfies The minus sign in the last two terms of the above quad- Iˉ2 = 1 , ratic form occurs because in 4-dimensional spacetime with 2 signature (+ ) we have I = (γ0γ1γ2γ3)(γ0γ1γ2γ3) = e Iˉ = Ieˉ , (181) − −− μ − μ = 1, and I†I = (γ3γ2γ1γ0)(γ0γ1γ2γ3) = 1. −In eq.-(185) the line element dxμdx −of the ordinary eμˉIˉ = Ieˉ μˉ . μ special or general relativity is replaced by the line element Then the Clifford numbers in Clifford space. A “square root” of such a generalized line
γμ = eμI,ˉ element is dX of eq.-(184). The latter object is a polyvector, (182) a differential of the coordinate polyvector field γμˉ = eμ 1 satisfy X = σ + xμγ + xμν γ γ +x ˜μIγ +σI ˜ , (186) γ γ = δ , μ 2 μ ∧ ν μ μ ∙ ν − μν γ γ = δ , (183) whose square is μˉ ∙ νˉ μˉνˉ γμ γμˉ = 0 . 2 2 μ 1 μν μ 2 ∙ X = σ + x xμ + x xμν x˜ x˜μ σ˜ . (187) | | 2 − − The set γA = (γμ, γμˉ) therefore spans the vector space of μ signature ( + + + ++). The polyvector X contains not only the vector part x γμ, − − − − − μν The examples above demonstrate how vector spaces of but also a scalar part σ, tensor part x γμ γν , pseudovector μ ∧ various signatures are obtained within a given set of poly- part x˜ Iγμ and pseudoscalar part σI˜ . Similarly for the vectors. Namely, vector spaces of different signature are differential dX. different subsets of polyvectors within the same Clifford When calculating the quadratic forms X 2 and dX 2 one | | | | algebra. In other words, vector spaces of different signature obtains in 4-dimensional spacetime with pseudo euclidean are different subspaces of C-space, i. e., different sections signature (+ ) the minus sign in front of the squares of − −− through C-space∗. the pseudovector and pseudoscalar terms. This is so, because This has important physical implications. We have argued in such a case the pseudoscalar unit square in flat spacetime 2 that physical quantities are polyvectors (Clifford numbers or is I = I†I = 1. In 4-dimensions I† = I regardless of the − Clifford aggregates). Physical space is then not simply a vec- signature. tor space (e.g., Minkowski space), but a space of polyvectors, Instead of Lorentz transformations — pseudo rotations μ μ called C-space, a pandimensional continuum of points, lines, in spacetime — which preserve x xμ and dx dxμ we have planes, volumes, etc., altogether. Minkowski space is then now more general rotations — rotations in C-space — which just a subspace with pseudo-Euclidean signature. Other sub- preserve X 2 and dX 2. | | | | spaces with other signatures also exist within the pandimen- sional continuum C and they all have physical significance. 7.2.2 C-space and conformal transformations If we describe a particle as moving in Minkowski spacetime V1,3 we consider only certain physical aspects of the object From (185) and (187) we see [25] that a subgroup of the Clif- considered. We have omitted its other physical properties like ford Group, or rotations in C-space is the group SO(4, 2). spin, charge, magnetic moment, etc. We can as well describe The transformations of the latter group rotate xμ, σ, σ˜, but μν μ the same object as moving in an Euclidean space V4. Again leave x and x˜ unchanged. Although according to our such a description would reflect only a part of the underlying assumption physics takes place in full C-space, it is very physical situation described by Clifford algebra. instructive to consider a subspace of C-space, that we shall call conformal space whose isometry group is SO(4, 2). ∗What we consider here should not be confused with the well known fact that Clifford algebras associated with vector spaces of different Coordinates can be given arbitrary symbols. Let us now signatures (p, q), with p + q = n, are not all isomorphic. use the symbol ημ instead of xμ, and η5,η6 instead of σ˜, σ. In
54 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
other words, instead of (xμ, σ,˜ σ) we write (ημ, η5, η6) ηa, Considering now a line element μ = 0, 1, 2, 3, a = 0, 1, 2, 3, 5, 6. The quadratic form reads≡ a μ dη dηa = dη dημ dκd , λ (199) a a b − η ηa = gabη η (188) a we find that on the cone η ηa = 0 it is with a 2 μ gab = diag(1, 1, 1, 1, 1, 1) (189) dη dη = κ dx dx (200) − − − − a μ being the diagonal metric of the flat 6-dimensional space, a even if κ is not constant. Under the transformation (194) we a subspace of C-space, parametrized by coordinates η . The have transformations which preserve the quadratic form (188) a a dη0 dηa0 = dη dηa , (201) belong to the group SO(4, 2). It is well known [38, 39] μ 2 μ that the latter group, when taken on the cone dx0 dxμ0 = ρ dx dxμ . (202)
a The last relation is a dilatation of the 4-dimensional line η ηa = 0 (190) element related to coordinates xμ. In a similar way also other is isomorphic to the 15-parameter group of conformal trans- transformations of the group SO(4, 2) that preserve (190) formations in 4-dimensional spacetime [40]. and (201) we can rewrite in terms of of the coordinates xμ. Let us consider first the rotations of η5 and η6 which So we obtain — besides dilations — translations, Lorentz leave coordinates ημ unchanged. The transformations that transformations, and special conformal transformations; al- leave (η5)2 + (η6)2 invariant are together they are called conformal transformations. This is a − well known old observation [38, 39] and we shall not discuss 5 5 6 η0 = η cosh α + η sinh α it further. What we wanted to point out here is that conformal (191) 6 5 6 group SO(4, 2) is a subgroup of the Clifford group. η0 = η sinh α + η cosh α , where α is a parameter of such pseudo rotations. 7.2.3 On the physical interpretation of the conformal Instead of the coordinates η5, η6 we can introduce [38, group SO(4, 2) 39] new coordinates κ, λ according to In order to understand the physical meaning of the transfor- μ μ κ = η5 η6 , mations (196) from the coordinates η to the coordinates x − (192) let us consider the following transformation in 6-dimensional 5 6 λ = η + η . space V6: xμ = κ 1ημ , In the new coordinates the quadratic form (188) reads − 1 α = κ− , (203) ηaη = ημη (η5)2 (η6)2 = ημη κλ . (193) − a μ μ 1 μ − − − Λ = λ κ− η η . − μ The transformation (191) becomes This is a transformation from the coordinates ηa = 1 = (ημ, κ, λ) to the new coordinates xa = (xμ, α, Λ). No extra κ0 = ρ− κ , (194) condition on coordinates, such as (190), is assumed now. If a a λ0 = ρλ , (195) we calculate the line element in the coordinates η and x , respectively, we find the the following relation [27] where ρ = eα. This is just a dilation of κ and the inverse dilation of . μ ν 2 μ ν λ dη dη gμν dκ dλ = α− (dx dx gμν dαdΛ) . (204) μ Let us now introduce new coordinates x ∗ − − We can interpret a transformation of coordinates pass- μ μ η = κx . (196) ively or actively. Geometric calculus clarifies significantly the meaning of passive and active transformations. Under Under the transformation (196) we have a passive transformation a vector remains the same, but μ μ its components and basis vector change. For a vector dη = η0 = η , (197) a = dη γa we have but μ μ (198) a a x0 = ρx , dη0 = dη0 γa0 = dη γa = dη (205) μ the latter transformation is dilatation of coordinates x . with a μ μ a ∂η0 b ∗These new coordinates x should not be confused with coordinate x dη0 = dη (206) used in section 2. ∂ηb
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a and The transformed line element dx dxa is physically dif- ∂ηb a (207) ferent from the original line element dη dηa by a factor γa0 = a γb . 2 2 ∂η0 α = κ− . 5 6 Since the vector is invariant, so it is its square: A rotation (191) in the plane (η , η ) i. e. the transforma- tion (194), (195) of (κ, λ) manifests in the new coordinates a a 2 a 2 a b a b a b x as a dilatation of the line element dx dx = κ− dη η : dη0 = dη0 γa0 dη0 γb0 = dη0 dη0 gab0 = dη dη gab . (208) a a a 2 a From (207) we read that the well known relation between dx0 dxa0 = ρ dx dxa . (216) new and old coordinates: a All this is true in the full space V6. On the cone η ηa = 0 ∂ηc ∂ηd μ a (209) we have Λ = λ κη ημ = 0, dΛ = 0 so that dx dxa = gab0 = a b gcd . μ − ∂η0 ∂η0 = dx dxμ and we reproduce the relations (202) which is a dilatation of the 4-dimensional line element. It can be Under an active transformation a vector changes. This interpreted either passively or actively. In general, the pseudo means that in a fixed basis the components of a vector rotations in V6, that is, the transformations of the 15-param- change: eter group SO(4, 2) when expressed in terms of coordinates a a a dη0 = dη0 γa (210) x , assume on the cone η ηa = 0 the form of the ordinary conformal transformations. They all can be given the active with a interpretation [27, 28]. a ∂η0 b dη0 = dη . (211) ∂ηb We started from the new paradigm that physical phe- nomena actually occur not in spacetime, but in a larger The transformed vector dη0 is different from the original space, the so called Clifford space or C-space which is a a vector dη = dη γa. For the square we find manifold associated with the Clifford algebra generated by the basis vectors γ of spacetime. An arbitrary element of ∂η a ∂η b μ d 2 d ad b 0 0 d cd d (212) Clifford algebra can be expanded in terms of the objects E , η0 = η0 η0 gab = c d η η gab , A ∂η ∂η A = 1, 2,..., 2D, which include, when D = 4, the scalar unit i. e., the transformed line element dη 2 is different from the 1, vectors γμ, bivectors γμ γν , pseudovectors Iγμ and the 0 ∧ original line element. pseudoscalar unit I γ5. C-space contains 6-dimensional ≡ Returning now to the coordinate transformation (203) subspace V6 spanned∗ by 1, γμ, and γ5. The metric of with the identification η a = xa, we can interpret eq.-(204) V6 has the signature (+ +). It is well known 0 − − − − passively or actively. that the rotations in V6, when taken on the conformal cone a In the passive interpretation the metric tensor and the η ηa = 0, are isomorphic to the non linear transformations of components dηa change under a transformation, so that in the conformal group in spacetime. Thus we have found out our particular case the relation (208) becomes that C-space contains — as a subspace — the 6-dimensional space V6 in which the conformal group acts linearly. From the a b 2 μ ν dx dx gab0 = α− (dx dx gμν dα dΛ) = physical point of view this is an important and, as far as we − (213) know, a novel finding, although it might look mathematically = dηadηbg = dημdην g dκ dλ ab μν − trivial. So far it has not been clear what could be a physical with interpretation of the 6 dimensional conformal space. Now we gμν 0 0 see that it is just a subspace of Clifford space. The two extra 2 1 g0 = α− 0 0 , dimensions, parameterized by κ and λ, are not the ordinary ab − 2 0 1 0 extra dimensions; they are coordinates of Clifford space C − 2 4 (214) of the 4-dimensional Minkowski spacetime V4. gμν 0 0 We take C-space seriously as an arena in which physics 1 gab = 0 0 2 . takes place. The theory is a very natural, although not trivial, 0 1 −0 − 2 extension of the special relativity in spacetime. In special In the above equation the same infinitesimal distance relativity the transformations that preserve the quadratic form a a squared is expressed in two different coordinates η or x . It is a well known observation that the generators L of SO (4, 2) a ∗ ab In active interpretation, only dη change, whilst the can be realized in terms of 1, γμ, and γ5. Lorentz generators are i 1 metric remains the same, so that the transformed element is Mμν = 4 [γμ, γν ], dilatations are generated by D = L65 = 2 γ5, − 1 − translations by Pμ = L5μ + L6μ = 2 γμ(1 iγ5) and the special conform- a b μ ν 1 − dx dx gab = dx dx gμν dα dΛ = al transformations by L5μ L6μ = γμ(1 + iγ5). This essentially means − 2 − (215) i 2 a b 2 μ ν that the generators are Lab = 4 [ea, eb] with ea = (γμ, γ5, 1), where care = κ− dη dη g = κ− (dη dη g dκ dλ) . − ab μν − must be taken to replace commutators [1, γ5] and [1, γμ] with 2γ5 and 2γμ.
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are given an active interpretation: they relate the objects or of polyvector-valued quantities and differential operators in the systems of reference in relative translational motion. C-space Analogously also the transformations that preserve the qua- N μ μ ν dratic form (185) or (187) in -space should be given an A = AN E = φ1 + Aμγ + Aμν γ γ + .... (217) C ∧ active interpretation. We have found that among such trans- The first component in the expansion is a scalar field; formations (rotations in C-space) there exist the transform- φ is the standard Maxwell field , the third component ations of the group SO(4, 2). Those transformations also Aμ Aμ should be given an active interpretation as the transforma- Aμν is a rank two antisymmetric tensor field. . . and the tions that relate different physical objects or reference frames. last component of the expansion is a pseudo-scalar. The fact Since in the ordinary relativity we do not impose any con- that a scalar and pseudo-scalar field appear very naturally straint on the coordinates of a freely moving object so we in the expansion of the C-space polyvector valued field AN suggests that one could attempt to identify the latter fields should not impose any constraint in C-space, or in the sub- with a dilaton-like and axion-like field, respectively. Once space V6. However, by using the projective coordinate trans- a again, in order to match units in the expansion (217), it formation (203), without any constraint such as η ηa = 0, we arrived at the relation (215) for the line elements. If requires the introduction of suitable powers of a length scale parameter, the Planck scale which is conveniently set to unity. in the coordinates ηa the line element is constant, then The differential operator is the generalized Dirac operator in the coordinates xa the line element is changing by a scale factor which, in general, depends on the evolution κ d = EM ∂ = 1∂ + γμ∂ + γμ γν ∂ + ... (218) parameter τ. The line element need not be one associated M σ xμ ∧ xμν between two events along a point particle’s worldline: it can the polyvector-valued indices M,N, . . . range from 1,2 ... 2D be between two arbitrary (space-like or time-like) events since a Clifford algebra in D-dim has 2D basis elements. The within an extended object. We may consider the line element generalized Maxwell field strength in C-space is ( distance squared) between two infinitesimally separated ≡ M N M N events within an extended object such that both events have F = dA = E ∂M (E AN ) = E E ∂M AN = the same coordinate label Λ so that dΛ = 0. Then the 6- 1 1 μ ν M N M N dimensional line element dx dx gμν dα dΛ becomes the = E ,E ∂M AN + [E ,E ]∂M AN = (219) μ ν − 2{ } 2 4-dimensional line element dx dx gμν and, because of 1 M N 1 M N (215) it changes with τ when κ does change. This means that = F(MN) E ,E + F[MN][E ,E ] , 2 { } 2 the object changes its size, it is moving dilatationally [27, 28]. We have thus arrived at a very far reaching observation that where one has decomposed the Field strength components the relativity in C-space implies scale changes of physical into a symmetric plus antisymmetric piece by simply writing objects as a result of free motion, without presence of any the Clifford geometric product of two polyvectors EM EN forces or such fields as assumed in Weyl theory. This was as the sum of an anticommutator plus a commutator piece advocated long time ago [27, 28], but without recurse to C- respectively, space. However, if we consider the full Clifford space C and 1 F = (∂ A + ∂ A ) , (220) not only the Minkowski spacetime section through C, then (MN) 2 M N N M we arrive at a more general dilatational motion [17] related μν μνα 0123 1 to the polyvector coordinates x , x and x σ˜ (also F[MN] = (∂M AN ∂N AM ) . (221) denoted s) as reviewed in section 3. ≡ 2 − Let the C-space Maxwell action (up to a numerical 7.3 C-space Maxwell Electrodynamics factor) be given in terms of the antisymmetric part of the field strength: Finally, in this section we will review and complement the [MN] proposal of ref. [75] to generalize Maxwell Electrodynamics I[A] = [DX]F[MN]F , (222) to C-spaces, namely, construct the Clifford algebra-valued Z extension of the Abelian field strength F = dA associated where [DX] is a C-space measure comprised of all the with ordinary vectors Aμ. Using Clifford algebraic methods (holographic) coordinates degrees of freedom we shall describe how to generalize Maxwell’s theory of Electrodynamics associated with ordinary point-charges to [DX] (dσ)(dx0dx1 ...)(dx01dx02 ...) ... ≡ a generalized Maxwell theory in Clifford spaces involving (223) ... (dx012...D) . extended charges and p-forms of arbitrary rank, not unlike the couplings of p-branes to antisymmetric tensor fields. Action (222) is invariant under the gauge transformations Based on the standard definition of the Abelian field strength F = dA we shall use the same definition in terms AM0 = AM + ∂M Λ . (224)
C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 57 Volume 1 PROGRESS IN PHYSICS April, 2005
The matter-field minimal coupling (interaction term) is: with respect to the XM variables:
2 N M M d XM dX AM dX = [DX]JM A , (225) (231) κ 2 = eF[MN] , Z Z dS dS where one has reabsorbed the coupling constant, the C-space where we have re-introduced the C-space charge e back into analog of the electric charge, within the expression for the the Lorentz force equation in C-space. A variation of the A field itself. Notice that this term (225) has the same form terms in the action w. r. t the AM field furnishes the following as the coupling of p-branes (whose world volume is (p +1)- equation of motion for the A fields: dimensional) to antisymmetric tensor fields of rank p + 1. [MN] N The open line integral in C-space of the matter-field ∂M F = J . (232) interaction term in the action is taken from the polyparticle’s proper time interval S ranging from to + and can be By taking derivatives on both sides of the last equation N recast via the Stokes law solely in terms−∞ of the antisymmetric∞ with respect to the X coordinate, one obtains due to the part of the field strength. This requires closing off the integ- symmetry condition of ∂M ∂N versus the antisymmetry of [MN] ration contour by a semi-circle that starts at S = + , goes F that ∞ [MN] N all the way to C-space infinity, and comes back to the point ∂N ∂M F = 0 = ∂N J = 0 , (233) S = . The field strength vanishes along the points ofthe −∞ semi-circle at infinity, and for this reason the net contribution which is precisely the continuity equation for the current. to the contour integral is given by the open-line integral. The continuity equation is essential to ensure that the M M Therefore, by rewriting the AM dX via the Stokes law matter-field coupling term of the action AM dX = relation, it yields = [DX]J M A is also gauge invariant, which can be read- R M ily verified after an integration by partsR and setting the M [MN] M N R AM dX = F[MN]dS = F[MN]X dX = boundary terms to zero: Z Z Z (226) M N M M = dSF[MN]X (dX /dS) , δ [DX]J AM = [DX]J ∂M Λ = Z Z Z (234) M where in order to go from the second term to the third = [DX](∂M J )Λ = 0. − term in the above equation we have integrated by parts and Z then used the Bianchi identity for the antisymmetric compo- Gauge invariance also ensures the conservation of the nent F[MN]. energy-momentum (via Noether’s theorem) defined in terms The integration by parts permits us to go from a C-space of the Lagrangian density variation. We refer to [75] for domain integral, represented by the Clifford-value hyper- further details. surface SMN , to a C-space boundary-line integral The gauge invariant C-space Maxwell action as given in eq.-(222) is in fact only a part of a more general action given 1 dSMN = (XM dXN XN dXM ) . (227) by the expression 2 − Z Z The pure matter terms in the action are given by the I[A] = [DX] F † F = [ X] < F †F > . (235) ∗ D scalar analog of the proper time integral spanned by the motion of Z Z a particle in spacetime: This action can also be written in terms of components, up to dimension-dependent numerical coefficients, as [75]: dXM dX κ dS = κ dS M , (228) r dS dS I[A] = [ X](F F (MN) + F F [MN]) . (236) Z Z D (MN) [MN] where κ is a parameter whose dimensions are massp+1 and Z S is the polyparticle proper time in C-space. For rigor, one should introduce the numerical coefficients The Lorentz force relation in C-space is directly obtained in front of the F terms, noticing that the symmetric combina- from a variation of tion should have a different dimension-dependent coefficient than the anti-symmetric combination since the former in- M N volves contractions of EM ,EN E ,E and the latter dSF[MN]X (dX /dS) , (229) ∗ M N M {N } { } Z contractions of [E ,E ]∗[EM ,EN ]. The latter action is strictly speaking not gauge invariant, and since it contains not only the antisymmetric but also the M κ dS = κ dX dXM (230) symmetric part of F . It is invariant under a restricted gauge Z Z p 58 C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces April, 2005 PROGRESS IN PHYSICS Volume 1
symmetry transformations. It is invariant (up to total deriva- sM , sN is the grade of EM and EN respectively. Even or tives) under infinitesimal gauge transformations provided the odd depending on the grade of the basis elements. (MN) symmetric part of F is divergence-free ∂M F = 0 [75]. One may generalize Maxwell’s theory to Born-Infeld This divergence-free condition has the same effects as if one nonlinear Electrodynamics in C-spacesbased on this exten- were fixing a gauge leaving a residual symmetry of restricted sion of Maxwell Electrodynamics in C-spaces and to couple gauge transformations such that the gauge symmetry pa- a C-space version of a Yang-Mills theory to C-space gravity, M rameter obeys the Laplace-like equation ∂M ∂ Λ = 0. Such a higher derivative gravity with torsion, this will be left for residual (restricted) symmetries are precisely those that leave a future publication. Clifford algebras have been used in invariant the divergence-free condition on the symmetric part the past [62] to study the Born-Infeld model in ordinary of F . Residual, restricted symmetries occur, for example, in spacetime and to write a nonlinear version of the Dirac eq- the light-cone gauge of p-brane actions leaving a residual uation. The natural incorporation of monopoles in Maxwell’s symmetry of volume-preserving diffs. They also occur in theory was investigated by [89] and a recent critical analysis string theory when the conformal gauge is chosen leaving of “unified” theories of gravity with electromagnetism has a residual symmetry under conformal reparametrizations; been presented by [90]. Most recently [22] has studied the i. e. the so-called Virasoro algebras whose symmetry trans- covariance of Maxwell’s theory from a Clifford algebraic formations are given by holomorphic and anti-holomorphic point of view. reparametrizations of the string world-sheet. This Laplace-like condition on the gauge parameter is also the one required such that the action in [75] is invariant 8 Concluding remarks under finite (restricted) gauge transformations since under such restricted finite transformations the Lagrangian changes We have presented a brief review of some of the most im- 2 by second-order terms of the form (∂M ∂N Λ) , which are portant features of the Extended Relativity theory in Clifford- total derivatives if, and only if, the gauge parameter is re- spaces (C-spaces). The “coordinates” X are non-commutat- M stricted to obey the analog of Laplace equation ∂M ∂ Λ = 0 ing Clifford-valued quantities which incorporate the lines, Therefore the action of eq-(233) is invariant under a areas, volumes, . . . degrees of freedom associated with the restricted gauge transformation which bears a resemblance collective particle, string, membrane, . . . dynamics under- to volume-preserving diffeomorphisms of the p-branes action lying the center-of-mass motion and holographic projections in the light-cone gauge. A lesson that we have from these of the p-loops onto the embedding target spacetime back- considerations is that the C-space Maxwell action written in grounds. C-space Relativity incorporates the idea of an in- the form (235) automatically contains a gauge fixing term. variant length, which upon quantization, should lead to the Analogous result for ordinary Maxwell field is known from notion of minimal Planck scale [23]. Other relevant features Hestenes work [1], although formulated in a slightly different are those of maximal acceleration [52], [49]; the invariance way, namely by directlty considering the field equations of Planck-areas under acceleration boosts; the resolution of without employing the action. ordering ambiguities in QFT; supersymmetry; holography It remains to be seen if this construction of C-space [119]; the emergence of higher derivative gravity with tor- generalized Maxwell Electrodynamics of p-forms can be sion; and the inclusion of variable dimensions/signatures that generalized to the non-Abelian case when we replace or- allows to study the dynamics of all (closed) p-branes, for all dinary derivatives by gauge-covariant ones: values of p, in one single unified footing, by starting with the C-space brane action constructed in this work. F = dA F = DA = (dA + A A). (237) The Conformal group construction presented in sect. 7, as → • a natural subgroup of the Clifford group in four-dimensions, For example, one could define the graded-symmetric needs to be generalized to other dimensions, in particular product EM EN based on the graded commutator of Super- to two dimensions where the Conformal group is infinite- algebras: • dimensional. Kinani [130] has shown that the Virasoro al- [A, B] = AB ( 1)sAsB BA, (238) gebra can be obtained from generalized Clifford algebras. − − The construction of area-preserving diffs algebras, like w ∞ s , s is the grade of A and B respectively. For bosons and su( ), from Clifford algebras remains an open problem. A B ∞ the grade is even and for fermions is odd. In this fashion Area-preserving diffs algebras are very important in the study the graded commutator captures both the anti-commutator of membranes and gravity since Higher-dim Gravity in of two fermions and the commutator of two bosons in one (m + n)-dim has been shown a while ago to be equivalent stroke. One may extend this graded bracket definition to the to a lower m-dim Yang-Mills-like gauge theory of diffs of graded structure present in Clifford algebras, and define an internal n-dim space [120] and that amounts to another explanation of the holographic principle behind the AdS/ E E = E E ( 1)sM sN E E , (239) CFT duality conjecture [121]. We have shown how C-space M • N M N − − N M
C. Castro and M. Pavsiˇ c.ˇ The Extended Relativity Theory in Clifford Spaces 59 Volume 1 PROGRESS IN PHYSICS April, 2005
Relativity involves scale changes in the sizes of physical for some time [95]. Wheeler’s spacetime foam at the Planck objects, in the absence of forces and Weyl’gauge field of scale may be the background source generation of Noise dilations. The introduction of scale-motion degrees of free- in the Parisi-Wu stochastic quantization [47] that is very dom has recently been implemented in the wavelet-based relevant in Number theory [100]. The pre-geometry cellular- regularization procedure of QFT by [87]. The connection networks approach of [107] and the quantum-topos views to Penrose’s Twistors program is another interesting project based on gravitational quantum causal sets, noncommutative worthy of investigation. topology and category theory [109], [110], [124] deserves The quantization and construction of QFTs in C-spaces a further study within the C-space Relativity framework, remains a very daunting task since it may involve the con- since the latter theory also invokes a Category point of view struction of QM in Noncommutative spacetimes [136], braid- to the notion of dimensions. C-space is a pandimensional ed Hopf quantum Clifford algebras [86], hypercomplex ex- continuum [14], [8]. Dimensions are topological invariants tensions of QM like quaternionic and octonionic QM [99], and, since the dimensions of the extended objects change in [97], [98], exceptional group extensions of the Standard C-space, topology-change is another ingredient that needs Model [85], hyper-matrices and hyper-determinants [88], to be addressed in C-space Relativity and which may shed multi-symplectic mechanics, the de Donde-Weyl formula- some light into the physical foundations of string/M theory tions of QFT [82], to cite a few, for example. The quantiza- [118]. It has been speculated that the universal symmetries tion program in C-spaces should share similar results as those of string theory [108] may be linked to Borcherds Vertex in Loop Quantum Gravity [111], in particular the minimal operator algebras (the Monstruous moonshine) that underline Planck areas of the expectation values of the area-operator. the deep interplay between Conformal Field Theories and Spacetime at the Planck scale may be discrete, fractal, Number theory. A lot remains to be done to bridge together fuzzy, noncommutative. . . The original Scale Relativity the- these numerous branches of physics and mathematics. Many ory in fractal spacetime [23] needs to be extended further surprises may lie ahead of us. For a most recent discussion on to incorporate the notion of fractal “manifolds”. A scale- the path towards a Clifford-Geometric Unified Field theory fractal calculus and a fractal-analysis construction that are of all forces see [138], [140]. The notion of a Generalized esential in building the notion of a fractal “manifold” has Supersymmetry in Clifford Superspaces as extensions of been initiated in the past years by [129]. It remains yet to be M,F theory algebras was recently advanced in [139]. proven that a scale-fractal calculus in fractal spacetimes is another realization of a Connes Noncommutative Geometry. Acknowledgements Fractal strings/branes and their spectrum have been studied by [104] that may require generalized Statistics beyond the We are indebted to M. Bowers and C. Handy at the CTSPS Boltzmann-Gibbs, Bose-Einstein and Fermi-Dirac, investi- (Atlanta) for support. The work of M. P. has been supported gated by [105], [103], among others. by the Ministry of Education, Science and Sport of Slovenia. Non-Archimedean geometry has been recognized long ago as the natural one operating at the minimal Planck scale and requires the use p-adic numbers instead of ordinary References numbers [101]. By implementing the small/large scale, ultraviolet/infrared duality principle associated with QFTs in 1. Hestenes D. Spacetime algebra. Gordon and Breach, New Noncommutative spaces, see [125] for a review, one would York, 1996. Hestenes D. and Sobcyk G. Clifford algebra expect an upper maximum scale [23] and a maximum tem- to geometric calculus. D. Reidel Publishing Company, perature [21] to be operating in Nature. Non-Archimedean Dordrecht, 1984. Cosmologies based on an upper scale has been investigated 2. Porteous I. R. Clifford algebras and the classical groups. by [94]. Cambridge University Press, 1995. An upper/lower scale can be accomodated simultane- 3. Baylis W. Electrodynamics, a modern geometric approach. ously and very naturally in the q-Gravity theory of [114], Boston, Birkhauser, 1999. [69] based on bicovariant quantum group extensions of the 4. Trayling G. and Baylis W. J. Phys., v. A34, 2001, 3309–3324. Poincare,´ Conformal group, where the q deformation param- Int. J. Mod. Phys., v. A16S1C, 2001, 909–912. eter could be equated to the quantity eΛ/L, such that both 5. Jancewicz B. Multivectors and clifford algebra in electro- Λ = 0 and L = , yield the same classical q = 1 limit. For dynamics. World Scientific, Singapore, 1989. ∞ a review of q-deformations of Clifford algebras and their 6. Clifford algebras and their applications in mathematical generalizations see [86], [128]. physics. Vol 1: Algebras and physics. Edited by R. Ablam- It was advocated long ago by Wheeler and others, that owicz, B. Fauser. Vol 2: Clifford analysis. Edited by J. Ryan, information theory [106], set theory and number theory, W. Sprosig. Birkhauser, Boston, 2000. may be the ultimate physical theory. The important role of 7. Lounesto P. Clifford algebras and spinors. Cambridge Uni- Clifford algebras in information theory have been known versity Press, Cambridge, 1997.
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Rational Numbers Distribution and Resonance
Kyril Dombrowski∗
This study solves a problem on the distribution of rational numbers along the number plane and number line. It is shown that the distribution is linked to resonance phenomena and also to stability of oscillating systems.
“God created numbers, all the rest has been created p 30 by Man. . . ”. With greatest esteem to Leopold Kronecker, one of the founders of the contemporary theory of numbers, it is impossible to agree with him in both the divine origin 25 of number and Man’s creation of mathematics. I propound herein the idea that numbers, their relations, and all mathem- 20 atics in general are objective realities of our world. A part of science is not only understanding things, but also studying 15 the relations that are objective realities in nature. In this work I am going to consider a problem concerning the distribution of rational numbers along the number line 10 and also in the number plane, and the relation of this distrib- ution to resonance phenomena and stability of oscillating 5 systems in low linear perturbations. Any oscillating process involving at least two interacting 1 oscillators is necessarily linked to abstract numbers — ratios 1 5 10 15 20 25 30 q between the oscillation periods. This fact displays a close Fig. 1: The lattice of numbers (Minkowski’s lattice). relationship between such sections of science as the physical theory of oscillations and the abstract theory of numbers. number of all possible pairs of natural numbers the points of As is well known, the rational numbers are distributed the lattice) approaches the limit on the number line everywhere compactly, so this problem Ra N 2 2 ∞ statement that a function of their distribution exists might lim = lim ϕ (m) = N N 2 N N 2 be thought false, as the case of prime numbers. But, as →∞ →∞ m=1 X we will see below, it is not false — a rational numbers 1 1 6 ∞ 1 − distribution function has an objective reality, manifest in = = = , ζ (2) π2 n2 numerous physical phenomena of Nature. This thesis will be- m=1 come clearer if we consider the “number lattice” introduced X where N is the number of rational numbers, Ra N 2 is by Minkowski (Fig. 1). Therein are given all points of coord- the number of rational numbers located inside the square inates p and q which are related to numerators and denomina- whose elements are of length equal to N, ϕ (m) is Euler’s tors, respectively. If we exclude all points of the Minkowski function, ζ (n) is Riemann’s zeta function, m and n are lattice with coordinates have a common divisor different natural numbers. In particular, we can conclude from this from unity, this plane will contain only “rational points” that when N < the average density of rational numbers p/q (the non-cancelled fractions). Their distribution in the located in the plane∞ is restricted to a very narrow interval of plane is defined by a sequence of numbers forming a rational numerical values. It is possible to this verify by very simple series (Fig. 1). calculations. This simplest drawing shows that rational numbers are di- To study the problem of what is common to the rational stributed inhomogeneously in the Minkowski number plane. number distribution and resonance phenomena it is necessary It is easy to see that this distribution is symmetric with respect to have a one-dimensional picture of the function Ra (x) on to the axis p =q. Numbers of columns (and rows) in intervals, the number line. In this problem, because the set of rational limited by this axis and one of the coordinate axes, are equal numbers is infinitely dense, we need to give a criterion for to Euler functions — the numbers less than m and relatively selecting a finite number of rational numbers which could prime with m. Therefore, if we expand the number lattice give an objective picture of their distribution on the number infinitely, the average density of rational numbers in the plane line. We can do this in two ways. First, we can study, for (the ratio between the number of rational numbers and the instance, the distribution of rational number rays, drawn
∗Translated from the Russian by D. Rabounski and S. J. Crothers. from the origin of coordinates in the Minkowski lattice. This
K. Dombrowski. Rational Numbers Distribution and Resonance 65 Volume 1 PROGRESS IN PHYSICS April, 2005
is a contemporary development of the method created by 1 150 mV xn xn 1 Klein [1]. Second, we can employ continued fractions, taking − − into account Khinchin’s remark that “continued fractions. . . in their pure form display properties of the numbers they represent” [2]. So we can employ the mathematical appar- 100 500 atus of continued fractions as a systematic ground in order to find an analogous result that had been previously obtained by a purely arithmetical way. We will use the second option because it is easier (alt- hough it is more difficult to imagine). So, let us plot points by 50 250 writing a single-term continued fraction 1/n (so these are the numbers 1/1, 1/2, 1/3, . . . ) inside an interval of unit length. We obtain thereby the best approximations of these numbers. 2a 2b This could be done inside every interval 1/ (n+1) < x < 1/n 0 0 1 1 √5 1/2 302 − by plotting points which are numerical values of a two-term 3 2 2 1 236 1000618414 √2 1 Frequency, Hz continued fraction − 1 n Fig. 2: The rational numbers distribution. 1 = . m + n mn + 1 amplifier (an active LC-filter having a frequency of 1kHz These points, according to the theory of continued fractions, and the quality Q = 17). The frequency at the input was are the best approximations of the numbers 1/n from the varied within the interval 200–1000 Hz through steps of left side.We then get the best approximations of the numbers 25 Hz. The average numerical value of the outgoing voltage 1/n from the right side, expressed by the fractions was measured for two different voltages of the incoming 1 n + 1 signal — 0.75V and 1.25V. 1 = , l, n, m = 1, 2, 3 .... The apexes of both functions shown in the diagrams l + 1 l (n + 1) + n 1+ n are located at the points plotted by the fractions 1/n. This fact is trivial, because both apexes are actually analogous to We will call the approximation obtained the first order Fourier-series expansions of white noise. Such experimental approximation (the second and third rank approximation in diagrams could be obtained in a purely theoretical way. Khinchin’s terminology). It is evident that every rational Much more interesting is the problem of the minimum point of k-th order obtained in this way has analogous se- numerical values of both functions. The classical theory of quences of the (k +1)-th rank and higher. Such sequences oscillations predicts that the minimum points should coincide fill the whole set of rational numbers. with the minimum amplitude of forced oscillations, while To consider the simplest cases of resonance it would be according to the theory of continued fractions the minimum enough to take the first order approximation, but to consider points should coincide with irrational numbers which, being numerous processes such as colour vision, musical harmony, the roots of the equation x2 px 1 = 0 for all p, are ap- or Bohr’s orbit distribution in atoms, requires a high order proximated by rational numbers± less− accurately than by other distribution function for rational numbers. numbers [2]. To obtain the function Ra (x) as a regular diagram we Direct calculations give the following numbers define this function (meaning the finite approximation order, the first order in this case) as a quantity in reverse tothe 1 √5 1 1 interval between the neighbouring rational points located on M1 = = ∓ = (0.6180339. . . )± , 1 + 1 2 the number line, where the points are plotted in the fashion 1+ 1 1+... of Khinchin, mentioned above. If the numerical values of the numbers l, m, n are limited, this interval is finite (see 1 √8 1 1 M2 = = ∓ = (0.4142135. . . )± , Fig. 2a). Such a drawing gives a possibility for estimating the 2 + 1 2 2+ 1 structure of rational number distribution along the number 2+... line. In Fig. 2a we consider the distribution structure of ...... rational numbers derived from a three-component continued 1 √n2 + 4 n fraction. For the purpose of comparison, Fig. 2b depicts a Mn = = ∓ . n + 1 2 voltage function dependent on the stimulating frequency in n+ 1 an oscillating contour (drawn in the same scale as that in n+... Fig. 2a). In this case an alternating signal frequency at a In other words, the first conclusion is that the distribution constant voltage was applied to the input of a resonance of rational numbers, represented by continued fractions with
66 K. Dombrowski. Rational Numbers Distribution and Resonance April, 2005 PROGRESS IN PHYSICS Volume 1
Table 1: Orbital radii of planets in the solar system in comparison between the orbital periods Tk/T0 or, alternatively, the ratios 2 with the calculated values of the radii Rk = √n + 4 n /2 ∓ between their functions (the average orbital radii Rk/R0) will be close to those numbers that correspond to the minima Rk(calc) Planet Real Rk Calculated Rk n of the rational numbers density on number line Rk(real) √n2 + 4 n Mercury 0.0744 0.0765 13 1.0282 Tk Rk − ; Mn = ∓ . Venus 0.1390 0.1401 7 1.0079 T0 R0 ≈ 2 Earth 0.1922 0.1926 −5 1.0021 The truth or falsity of this can be decided by using Table 1 Mars 0.2929 0.3028 −3 1.0338 Asteroids 0.6180 0.6180 −1 1.0000 and Table 2. Jupiter 1.0000 1.0000 −0 1.0000 As a matter of fact, all that has been said on the distribut- Saturn 1.8334 1.6180 1 0.8825 ion of rational numbers on a unit interval could be extrapol- Uranus 3.6883 3.3028 3 0.8955 ated for the entire number line (proceeding from the above Neptune 5.7774 5.1926 5 0.8988 mentioned concept). Pluto 7.6398 7.1401 7 0.9346 All that has been said gives a possibility to formulate the next conclusions: Table 2: Orbital periods of planets in the solar system 1. Rational numbers having limited numerator and de- in comparison with the calculated values of the periods nominator are distributed inhomogeneously along the 2 Tk = √n + 4 n /2 ∓ number line; 2. Oscillating systems, having a peculiarity to change Tk(calc) Planet Real Tk Calculated Tk n their own parameters because of interactions inside the Tk(real) systems, have a tendency to reach a stable state where Mercury 0.0203 0.0203 49 1.0000 the separate oscillators frequencies are interrelated by Venus 0.0519 0.0524 −19 1.0096 specific numbers — minima of the rational number − Earth 0.0843 0.0828 12 0.9822 density on number line. Mars 0.1586 0.1623 − 6 1.0233 Asteroids 0.4877 0.4142 −2 0.8493 − Jupiter 1.0000 1.0000 0 1.0000 References Saturn 2.4834 2.4142 2 0.9721 Uranus 7.0827 7.1378 7 1.0077 1. Klein F. Elementarmathematik vom hoheren¨ Standpunkte aus Neptune 13.8922 14.0711 13 1.0129 erster Band. Dritte Auflage, Springer Verlag, Berlin, 1924. Pluto 21.1166 21.0475 21 0.9967 2. Khinchin A. Ya. Continued fractions. Nauka, Moscow, 1978. Note: Here the measurement units are the orbital radius and 3. Dombrowski K. I. Bulletin of Soviet Atron. Geodesical Society, period of Jupiter. For asteroids the overall average orbit is taken, 1956, No. 17 (24), 46–50. its radius 3.215 astronomical units and period 5.75 years are the average values between asteroids. a limited number of elements, takes its minimum density at the points of a unit interval on number line as shown by the aforementioned numbers. The second conclusion is that if these numbers express ratios between interacting frequencies, the amplitude of the forced oscillations takes its minimum numerical value. It is evident that an oscillating system, where the oscil- lation parameters undergo changes due to interactions inside the system, will be maximally stable in that case where the forced oscillation amplitude will be a minimum. The simplest verification of this thesis is given bythe solar system. As we know it Laplace’s classic works, the whole solar system (the planet orbits on the average) are stable under periodic gravitational perturbations only if the ratios between the orbital parameters are expressed by ir- rational numbers. If we will take this problem forward, proceeding from the viewpoint proposed above, the ratios
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On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy
Stephen J. Crothers Australian Pacific College, Lower Ground, 189 Kent St., Sydney, 2000, Australia E-mail: [email protected]
The general solution to Einstein’s vacuum field equations for the point-mass in all its configurations must be determined in such a way as to provide a means bywhich an infinite sequence of particular solutions can be readily constructed. Itisfrom such a solution that the underlying geometry of Einstein’s universe can be rightly explored. I report here on the determination of the general solution and its consequences for the theoretical basis of relativistic degeneracy, i. e. gravitational collapse and the black hole.
1 Introduction required solution must satisfy must be established. Abrams [9] has determined these conditions. I obtain them by other A serious misconception prevails that the so-called “Schwarz- arguments, and therefrom construct the general solution, schild solution” is a solution for the vacuum field. Not only from which the original Schwarzschild solution, the Droste/ is this incorrect, it is not even Schwarzschild’s solution. Weyl solution, and the Brillouin solution all arise quite nat- The aforesaid solution was obtained by David Hilbert [1], urally. It will be evident that the black hole is theoretically a full year after Karl Schwarzschild [2] obtained his original unsound. Indeed, it never arose in the solutions of Schwarz- solution. Moreover, Hilbert’s metric is a corruption of the schild, Droste and Weyl, and Brillouin. It comes solely from solution first found by Johannes Droste [3], and subsequently the mathematically inadmissible assumptions conventionally by Hermann Weyl [4] by a different method. imposed upon the Hilbert metric. The orthodox concepts of gravitational collapse and the I provide herein a derivation of the general solution black hole owe their existence to a confusion as to the for the simple point-mass and briefly discuss its geometry. true nature of the r-parameter in the metric tensor for the Although I have obtained the complete solution up to the gravitational field. rotating point-charge I reserve its derivation to a subsequent The error in the conventional analysis of Hilbert’s solu- paper and similarly a full discussion of the geometry to a tion is twofold in that two tacit and invalid assumptions are third paper. However, I include the expression for the overall made: general solution as a prelude to my following papers. (a) r is a coordinate and radius (of some kind) in the gravitational field; 2 The general solution for the simple point-mass and (b) The regions 0 < r < α = 2m and α < r < are valid. its basic geometry ∞ Contrary to the conventional analysis the nature and range or the r-parameter must be determined by rigorous A general metric for the static, time-symmetric, mathematical means, not by mere assumption, tacit or other- cento-symmetric configuration of energy or matter in quasi- wise. When the required mathematical rigour is applied it Cartesian coordinates is, is revealed that r0 = α denotes a point, not a 2-sphere, ds2 = L(r)dt2 M(r)(dx2 + dy2 + dz2) and that 0 < r < α is undefined on the Hilbert metric. The − − N(r)(xdx + ydy + zdz)2, consequence of this is that gravitational collapse, if it occurs − (1) in Nature at all, cannot produce a relativistic black hole r = x2 + y2 + z2 , under any circumstances. Since the Michell-Laplace dark body is not a black hole either, there is no theoretical basis where, t,pL, M, N are analytic functions such that, for it whatsoever. Furthermore, the conventional conception ∀ of gravitational collapse is demonstrably false. L, M, N > 0 . (2) The sought for general solution must not only result in a In polar coordinates (1) becomes, means for construction of an infinite sequence of particular solutions, it must also naturally produce the solutions due ds2 = A(r)dt2 B(r)dr2 C(r)(dθ2 + sin2 θdϕ2), (3) to Schwarzschild, Droste and Weyl, and M. Brillouin [5]. To − − obtain the general solution the general conditions that the where analytic A, B, C > 0 owing to (2).
68 S J. Crothers. On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy April, 2005 PROGRESS IN PHYSICS Volume 1
Transform (3) by setting Rp, in keeping with the geometrical relations on (8), is now given by, r r∗ = C(r) , (4) Rp = √ g11dr , (11) r − then p Z 0 2 2 2 ds = A∗(r∗)dt B∗(r∗)dr∗ where from (7), − − (5) 1 2 2 2 2 α − [C (r)]2 r∗ (dθ + sin θdϕ ) , g = 1 0 . (12) − − 11 − 4C(r) from which one obtains in the usual way, C(r)! Equation (11) with (12)p gives the mapping of d from the 2 r∗ α 2 r∗ 2 ds = − dt dr∗ flat spacetime of Special Relativity into the curved spacetime r − r α − ∗ ∗ − (6) of General Relativity, thus, 2 2 2 2 r∗ (dθ + sin θdϕ ) . − √C C0 Substituting (4) gives Rp(r) = dr = s√C α 2√C Z − 2 √C α √C C0 ds2 = − dt2 dr2 = C(r) C(r) α + √C − √C α 4C − − ! ! (7) r (13) − p p C(dθ2 + sin2 θdϕ2) . C(r) + C(r) α − + α ln − , q Kq Thus, (7) is a general metric in terms of one unknown p p function C(r). The following arguments are coordinate in- dependent since C(r) in (7) is an arbitrary function. K = const . The general metric for Special Relativity is, The relationship between r and Rp is ds2 = dt2 dr2 r2 dθ2 + sin2 dϕ2 , (8) − − r r0 Rp 0 , → ⇒ → and the radial distance (the proper distance) between two so from (13) it follows, points is, r 2 r r0 C(r0) = α ,K = √α . d = dr = r r0 . (9) → ⇒ r − Z 0 So (13) becomes, Let a test particle be located at each of the points r0 and r > r0 (owing to the isotropy of space there is no loss of Rp(r) = C(r) C(r) α + generality in taking r > r0 > 0). Then by (9) the distance − r between them is given by p p (14) C(r) + C(r) α + α ln − . d = r r0 , q √qα − p p and if r = 0, d r in which case the distance from r = 0 0 ≡ 0 is the same as the radius (the curvature radius) of a great Therefore (7) is singular only at r = r0, where C(r0) = 2 circle, the circumference χ of which is from (8), = α and g00 = 0 r0, irrespective of the value of r0. 2 ∀ C(r0) = α emphasizes the true meaning of α, viz., α is χ = 2π√r2 = 2πr . (10) a scalar invariant which fixes the spacetime for the point- mass from an infinite number of mathematically possible In other words, the curvature radius and the proper radius forms, as pointed out by Abrams. Moreover, α embodies are identical, owing to the pseudo-Euclidean nature of (8). the effective gravitational mass of the source of the field, Furthermore, d gives the radius of a sphere centred at the and fixes a boundary to an otherwise incomplete spacetime. point r0. Let the test particle at r0 acquire mass. This pro- Furthermore, one can see from (13) and (14) that r0 is duces a gravitational field centred at the point r0 > 0. The arbitrary, i. e. the point-mass can be located at any point geometrical relations between the components of the metric and its location has no intrinsic meaning. Furthermore, the tensor of General Relativity must be precisely the same condition g00 = 0 is clearly equivalent to the boundary con- in the metric of Special Relativity. Therefore the distance dition r r R 0, from which it follows that g = 0 → 0 ⇒ p → 00 between r0 and r > r0 is no longer given by (9) and the is the end result of gravitational collapse. There exists no curvature radius no longer by (10). Indeed, the proper radius value of r making g11 = 0.
S J. Crothers. On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy 69 Volume 1 PROGRESS IN PHYSICS April, 2005
If C0 = 0 for r > r0 the structure of (7) is destroyed: where n and r0 are arbitrary. Therefore with r0 arbitrary, (17) g = 0 for r > r B(r) = 0 for r > r in violation of (3). reduces to the metric of Special Relativity when α = 2m = 0. 11 0 ⇒ 0 Therefore C0 = 0. For (7) to be spatially asymptotically flat, From (17), with r0 = 0 and n taking integer values, the 6 following infinite sequence obtains: C(r) lim = 1 . (15) 2 r 2 C1(r) = (r + α) (Brillouin’s solution) →∞ r r0 − 2 2 C2(r) = (r + α ) 2 Since C(r) must behave like (r ro) and make (7) 3 3 2 − C3(r) = (r + α ) 3 (Schwarzschild’s solution) singular only at r = r0, C(r) must be a strictly monotonically 4 4 1 increasing function. Then by virtue of (15) and the fact that C4(r) = (r + α ) 2 , etc. C = 0, it follows that C > 0 for r > r . Thus the necessary 0 0 0 Hilbert’s solution is rightly obtained when r0 = α, i. e. conditions6 that must be imposed upon C(r) to render a when r0 = α and the values of n take integers, the infinite solution to (3) are: sequence of particular solutions is then given by, 1. C (r) > 0 for r > r ; 2 0 0 C1(r) = r [Droste/Weyl/(Hilbert) solution] C(r) 2 2 2. lim = 1; C2(r) = (r α) + α , r 2 − → ∞ r r0 2 − C (r) = (r α)3 + α3 3 , 3. C(r ) = α2. 3 − 0 1 4 4 2 I call the foregoing the Metric Conditions of Abrams for C4(r) = (r α) + α , etc. − the point-mass (MCA) since when r = 0 they are precisely ijkm 0 The curvature f = R Rijkm is finite everywhere, in- the conditions he determined by his use of (3) and the field cluding r = r0. Indeed, for metric (17) the Kretschmann equations. In addition to MCA any admissible function C(r) scalar is, must reduce (7) to the metric of Special Relativity when 12α2 12α2 α = 2m = 0. f = 3 = 6 . (18) Cn n n The invalid conventional assumptions that 0 < r < α and r r + αn − 0 that r is a radius of sorts in the gravitational field lead h i to the incorrect conclusion that r = α is a 2-sphere in the Gravitational collapse does not produce a curvature sin- gravitational field of the point-mass. The quantity r = α does gularity in the gravitational field of the point-mass. The scalar 12 not describe a 2-sphere; it does not yield a Schwarzschild invariance of f(r0) = α4 is evident from (18). sphere; it is actually a point. Stavroulakis [10, 8, 9] has also All the particular solutions of (17) are inextendible, since remarked upon the true nature of the r-parameter (coordinate the singularity when r = r0 is quasiregular, irrespective of the radius). Since MCA must be satisfied, admissible systems of values of n and r0. Indeed, the circumference χ of a great coordinates are restricted to a particular (infinite) class. To circle becomes, satisfy MCA, and therefore (3), and (7), the form that C(r) χ = 2π C(r). (19) can take must be restricted to, Then the ratio p 2 n n n χ Cn(r) = (r r0) + α , (16) lim , (20) − r r Rp → ∞ → 0 + r0 ( −), n , shows that is a quasiregular singularity and can- ∈ < − < ∈ < Rp(r0) 0 not be extended. ≡ where n and r are arbitrary. I call equations (16) Schwarz- 0 Equation (19) shows that χ = 2πα is also a scalar invar- schild forms. The value of n in (16) fixes a set of coordinates, iant for the point-mass. and the infinitude of such reflects the fact that nosetof coordinates is privileged in General Relativity. It is plain from the foregoing that the Kruskal-Szekeres The general solution for the simple point-mass is there- extension is meaningless, that the “Schwarzschild radius” is fore, meaningless, that the orthodox conception of gravitational 2 collapse is incorrect, and that the black hole is not consistent √C α √C C0 ds2 = n − dt2 n n dr2 at all with General Relativity. All arise wholely from a √Cn − √Cn α 4Cn − (17) − bungled analysis of Hilbert’s solution. C (dθ2 + sin2 θdϕ2) , − n 2 3 Implications for gravitational collapse n n n + Cn(r) = (r r0) + α , n , − ∈ < As is well known the gravitational potential Φ for an arbitrary r ( −), 0 ∈ < − < metric is r < r < , g = (1 Φ)2 , (21) 0 ∞ 00 −
70 S J. Crothers. On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy April, 2005 PROGRESS IN PHYSICS Volume 1
from which it is concluded that gravitational collapse occurs Then α at Φ = 1. Physically, the conventional process of collapse as r , g00 1 , → ∞ → − r r involves Newtonian gravitation down to the so-called “grav- − 0 itational radius”. Far from the source, the alleged weak field and Newton is recovered at infinity. 1 potential is, According to (23), at r = r0, g00 = 0 and Φ = 2 . The m weak field potential approaches a finite maximum of 1 Φ = , 2 r 1 2 (i. e. 2 c ), in contrast to Newton’s potential. The conven- and so α tional concept of gravitational collapse at rs = α is therefore g00 = 1 , (22) meaningless. − r Similarly, it is unreasonable to expect Kepler’s 3rd Law α = 2m . to be unaffected by general relativity, contrary to the con- The scalar α is conventionally called the “gravitational ventional analysis. Consider the Lagrangian, radius”, or the “Schwarzschild radius”, or the “event hori- zon”. However, as I have shown, neither α nor the coordinate 1 α dt 2 L = 1 radius r are radii in the gravitational field. In the case of the 2 " − √Cn dτ # − Hilbert metric, r = α is a point, not a 2-sphere. It is the 0 1 2 1 α − d√C location of the point-mass. In consequence of this g00 = 0 is 1 n the end result of gravitational collapse. It therefore follows − 2 − √C dτ − " n # that in the vacuum field, 1 dθ 2 dϕ 2 (24) C + sin2 θ , 0 < g < 1 , 1 < g < , − 2 n dτ dτ 00 | 11| ∞ " !# 2 α < C(r) . n n n + Cn(r) = r r0 + α , n , 2 − ∈ < In the case of the Hilbertp metric, C(r) = r , so h i r ( −), r < r < , 0 ∈ < − < 0 ∞ 0 < g00 < 1, 1 < g11 < , | | ∞ where τ is the proper time. α < r . Restricting motion, without loss of generality, to the π In the case of Schwarzschild’s metric we have C(r) = equatorial plane, θ = 2 , the Euler-Lagrange equations for 2 (24) are, = r3 + α3 3 , so 1 2 2 α − d √Cn α dt 0 < g00 < 1, 1 < g11 < , 1 2 + | | ∞ − √C dτ 2Cn dτ − n (25) 0 < r . 2 2 2 α − α d√Cn dϕ 1 Cn =0, It is unreasonable to expect the weak field potential − − √C 2C dτ − dτ n n function to be strictly Newtonian. Only in the infinitely p far field is Newton’s potential function to be recovered. α dt 1 = const = k , (26) Consequently, the conventional weak field expression (22) − √Cn dτ cannot be admitted with the conventional interpretation there- of. The correct potential function must contain the arbitrary dϕ Cn = const = h , (27) location of the point-mass. From (21), dτ 2 μ ν α and ds = gμν dx dx becomes, Φ = 1 √g = 1 1 , − 00 − − s C(r) 2 α dt so in the weak far field, 1 p −√C dτ − n (28) 1 2 2 α m α − d√C dϕ Φ 1 1 = , 1 n C = 1 . ≈ − − 2√C √C n − −√Cn dτ − dτ and so α α Using the foregoing equations it readily follows that the g00 = 1 = 1 1 , (23) angular velocity is, − C(r) − (r r )n + αn n − 0 α p ω = 3 . (29) + 2 r ( −), n . 2C 0 ∈ < − < ∈ < s n
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Then, This is a time dependent metric which does not have any 1 lim ω = (30) relationship to the original static problem. It does not extend r r α√2 (17) at all, as also noted by Brillouin in the particular solution → 0 is a scalar invariant which shows that the angular velocity given by him. Equation (34) is meaningless. approaches a finite limit, in contrast to Newton’s theory It is noteworthy that Hagihara [11] has shown that all where it becomes unbounded. Schwarzschild obtained this geodesics that do not run into the Hilbert boundary at r0 = α result for his particular solution. Equation (29) is the General are complete. His result is easily extended to any r0 > 0 Relativistic modification of Kepler’s 3rd Law. in (17). For a falling particle in a true Schwarzschild field, The correct conclusion is that gravitational collapse ter- minates at the point-mass without the formation of a black α hole in all general relativistic circumstances. dτ = √g00 dt = 1 dt . s − C(r) 4 Generalization of the vacuum solution for charge and p Therefore, as a neutral test particle approaches the field angular momentum source at r along a radial geodesic, dτ 0. Thus, according 0 → to an external observer, it takes an infinite amount of coor- The foregoing analysis can be readily extended to include the dinate time for a test particle to reach the source. Time stops charged and rotating point-mass. In similar fashion it follows at the Schwarzschild point-mass. The conventional concepts that the Reissner-Nordstrom, Carter, Graves-Brill, Kerr, and of the Schwarzschild sphere and its interior are meaningless. Kerr-Newman black holes are all inconsistent with General Doughty [10] has shown that the acceleration of a test Relativity. particle approaching the point-mass along a radial geodesic In a subsequent paper I shall derive the following overall is given by, general solution for the point-mass when Λ = 0, √ g g11 g − 11 − | 00,1| a = . (31) 2 Δ 2 2 2g00 ds = dt a sin θdϕ ρ2 − − By (17), 2 2 2 sin θ 2 ρ C α C + a2 dϕ adt n0 dr2 ρ2dθ2 , a = 1 . 2 n 3 2 − ρ − − Δ 4Cn − 2C 4 √C α − 2 n n n Clearly, as r r0, a , independently of the value of Cn(r) = r r0 + β , r0 ( −) , → →2 ∞ − ∈ < − < r0. In the case of C(r) = r , where r0 = α, h i + L 2 2 2 α n , a = , ρ = Cn + a cos θ , a = 3 , (32) ∈ < m 2 √ 2r r α 2 2 − Δ = Cn α Cn + q + a , so a as r r0 = α. − Applying→ ∞ (31)→ to the Kruskal-Szekeres extension gives p rise to the absurdity of an infinite acceleration at r = α β = m + m2 q2 a2 cos2 θ, a2 + q2 < m2, − − where it is conventionally claimed that there is no matter p r < r < . and no singularity. It is plainly evident that gravitational 0 ∞ collapse terminates at a Schwarzschild simple point-mass, The different configurations for the point-mass are easily not in a black hole. Also, one can readily see that the alleged extracted from this set of equations by the setting of the interchange of the spatial and time coordinates “inside” the values of the parameters in the obvious way. “Schwarzschild sphere” is nonsensical. To amplify this, in (17), suppose C(r) < α, then Dedication 1 pα α − C 2 ds2 = 1 dt2 + 1 0 dr2 I dedicate this paper to the memory of Dr. Leonard S. − √ − √ − 4C − C C (33) Abrams: (27 Nov. 1924 — 28 Dec. 2001). C dθ2 + sin2 dϕ2 . − Epilogue Let r = t and t = r, then 1 α √C − C˙ 2 α √C My interest in the problem of the black hole was aroused by ds2 = e − e dt2 − dr2 coming across the papers of the American physicist Leonard √C 4C − √C − (34) S. Abrams, and subsequently to the original papers of C(et) dθ2 + sin2 dϕ2 . Schwarzschild, Droste, Weyl, Hilbert, and Brillouin. I was − e e 72 S J. Crothers. On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy April, 2005 PROGRESS IN PHYSICS Volume 1
drawn to the logic of Abrams’ approach in his determination 7. Stavroulakis N. A statical smooth extension of Schwarzschild’s of the required metric in terms of a single generalised func- metric. Lettere al Nuovo Cimento, 1974, v. 11, 8. tion and the conditions that this function must satisfy to 8. Stavroulakis N. On the Principles of General Relativity and render a solution for the point-mass. It was not until I read the SΘ(4)-invariant metrics. Proc. 3rd Panhellenic Congr. Abrams that I became aware of the startling facts that the Geometry, Athens, 1997, 169. “Schwarzschild solution” is not due to Schwarzschild, that 9. Stavroulakis N. On a paper by J. Smoller and B. Temple. Schwarzschild did not predict the black hole and made none Annales de la Fondation Louis de Broglie, 2002, v. 27, 3 (see of the claims about black holes that are invariably attributed also in www.geocities.com/theometria/Stavroulakis-1.pdf). to him in the textbooks and almost invariably in the literature. 10. Doughty N. Am. J. Phys., 1981, v. 49, 720. These facts alone give cause for disquiet and reading of the 11. Hagihara Y. Jpn. J. Astron. Geophys., 1931, v. 8, 67. original papers gives cause for serious concern about how modern science is reported. Dr. Leonard S. Abrams was born in Chicago in 1924 and died on December 28, 2001, in Los Angeles at the age of 77. He received a B. S. in Mathematics from the California Institute of Technology and a Ph. D. in physics from the University of California at Los Angeles at the age of 45. He spent almost all of his career working in the private sector, although he taught at a variety of institutions including California State University at Dominguez Hills and at the University of Southern California. He was a pioneer in applying game theory to business problems and was an expert in noise theory, but his first love always was general relativity. His principle theoretical contributions focused on non-black hole solutions to Einstein’s equations and on the inextendability of the “Schwarzschild” solution. Dr. Abrams is survived by his wife and two children. Dr. Abrams encountered great resistance to publication of his work on General Relativity. Nonetheless he continued with his work and managed to publish several important papers despite the obstacles placed in his way by the main- stream authorities. I extend my thanks to Diana Abrams for providing me with information about her late husband.
References
1. Hilbert D. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl., 1917, 53, (see this Ref. in arXiv: physics/0310104). 2. Schwarzschild K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 189 (see this item also in arXiv: physics/9905030. 3. Droste J. The field of a single centre in Einstein’s theory of gravitation, and the motion of a particle in that field. Ned. Acad. Wet., S. A., 1917, v. 19, 197 (see also in www.geocities.com/ theometria/Droste.pdf). 4. Weyl H. Zur Gravitationstheorie. Ann. Phys. (Leipzig), 1917, v. 54, 117. 5. Brillouin M. The singular points of Einstein’s Universe. Journ. Phys. Radium, 1923, v. 23, 43 (see also in arXiv: physics/ 0002009). 6. Abrams L. S. Black holes: the legacy of Hilbert’s error. Can. J. Phys., 1989, v. 67, 919 (see also in arXiv: gr-qc/0102055).
S J. Crothers. On the General Solution to Einstein’s Vacuum Field and Its Implications for Relativistic Degeneracy 73 Volume 1 PROGRESS IN PHYSICS April, 2005
On the Ramifications of the Schwarzschild Space-Time Metric
Stephen J. Crothers Australian Pacific College, Lower Ground, 189 Kent St., Sydney, 2000, Australia E-mail: [email protected]
In a previous paper I derived the general solution for the simple point-mass in a true Schwarzschild space. I extend that solution to the point-charge, the rotating point- mass, and the rotating point-charge, culminating in a single expression for the general solution for the point-mass in all its configurations when Λ = 0. The general exact solution is proved regular everywhere except at the arbitrary location of the source of the gravitational field. In no case does the black hole manifest. The conventional solutions giving rise to various black holes are shown to be inconsistent with General Relativity.
1 Introduction where Cn(r) satisfies the Metric conditions of Abrams (MCA) [2]∗ for the simple point-mass, In a previous paper [1] I showed that the general solution of the vacuum field for the simple point-mass is regular 1. Cn0 (r) > 0, r > r0; everywhere except at the arbitrary location of the source C (r) 2. lim n = 1; of the field, r = r , r ( ), where there is a r 2 0 0 − →∞ r r0 quasiregular singularity. I extend∈ < herein − < the general solution − 3. C (r ) = α2 . to the rotating and charged configurations of the point-mass n 0 and show that they too are regular everywhere except at The Reissner-Nordstrom [3] solution is, r = r0, obviating the formation of the Reissner-Nordstrom, 1 Kerr, and Kerr-Newman black holes. Consequently, there is α q2 α q2 − ds2 = 1 + dt2 1 + dr2 no basis in General Relativity for the black hole. − r r2 − − r r2 − (3) The sought for complete solution for the point-mass r2(dθ2 + sin2 θdϕ2) , must reduce to the general solution for the simple point- − mass in a natural way, give rise to an infinite sequence 2 which is conventionally taken to be valid for all q . It is of particular solutions in each particular configuration, and m2 contain a scalar invariant which embodies all the factors that also alleged that (3) can be extended down to r = 0, giving contribute to the effective gravitational mass of the field’s rise to the so-called Reissner-Nordstrom black hole. These source for the respective configurations. conventional allegations are demonstrably false. The conventional analysis simply looks at (3) and makes two mathematically invalid assumptions, viz., 2 The vacuum field of the point-charge 1. The parameter r is a radius of some kind in the grav- The general metric, in polar coordinates, for the vacuum field itational field; is, in relativistic units, 2. r down to r = 0 is valid. ds2 = A(r)dt2 B(r)dr2 C(r)(dθ2 + sin2 θdϕ2) , (1) The nature and range of the r-parameter must be estab- − − lished by mathematical rigour, not by mere assumption. where analytic A, B, C > 0. The general solution to (1) for Transform (1) by the substitution the simple point-mass is, r∗ = C(r). (4) 2 (√Cn α) √Cn C0 ds2 = − dt2 n dr2 p √C − (√C α) 4C − ∗Abrams’ equation (A.1) should read: n n n (2) − 1 C 2 A C 2 2 2 8πT 1 = − + 0 + 0 0 = 0 , C (dθ + sin θdϕ ) , 1 2 − n − C 4BC 2ABC and his equation (A.6), 2 n n n C (r) = r r + α , α = 2m , r ( −) , 2C00 n 0 0 [ln (ABC)]0 = 0 . − ∈ < − < C − h i 0 n + , r < r < , The errors are apparently escapees from the proof reading. ∈ < 0 ∞
74 S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric April, 2005 PROGRESS IN PHYSICS Volume 1
Equation (4) carries (1) into Then by (9),
2 2 2 2 2 2 2 2 2 2 2 ds = A∗(r∗)dt B∗(r∗)dr∗ r∗ (dθ + sin θdϕ ). (5) K = m q , q < m . (11) − − − p Clearly, r0 is the lower bound on r. Using (5) to determine the Maxwell stress-energy tensor, Putting (11) into (9) gives, and substituting the latter into the Einstein-Maxwell field equations in the usual way, yields, R (r) = C(r) α C(r) + q2 + p − α q2 q 2 2 p 2 (12) ds = 1 + 2 dt C(r) m+ C(r) α C(r)+q − r∗ r∗ − + m ln − − . q 2 2 1 (6) p m q p α q2 − − 1 + dr 2 r 2(dθ2 + sin2 θdϕ2) . 2 ∗ ∗ p − − r∗ r∗ − Equation (7) is therefore singular only when r = r0 in which case g00 = 0. Hence, the condition r r0 Rp 0 Substituting (4) into (6), is equivalent to r = r g = 0. → ⇒ → 0 ⇒ 00 If C0 = 0 the structure of (7) is destroyed, since g11 = 0 2 2 1 ∀ 2 α q 2 α q − r > r B(r) = 0 r > r in violation of (1). Therefore ds = 1 + dt 1 + 0 ⇒ ∀ 0 − √C C − − √C C × C0(r) = 0 for r > r . (7) 0 For6 (7) to be asymptotically flat, C 2 0 dr2 C(dθ2 + sin2 θdϕ2) . × 4C − C(r) r 2 1 . (13) → ∞ ⇒ r r → The proper radius Rp on (1) is, − 0 Therefore, C(r) Rp(r) = B(r) dr . (8) lim = 1 . (14) r 2 →∞ r r0 Z p − The parameter r therefore does not lie in the spacetime 2 Since C(r) behaves like r r0 , must make (7) sin- M of the point-charge. − q gular only at r = r0, and C0(r) = 0 for r > r0, C(r) is strictly Taking B(r) from (7) into (8) gives the proper distance 6 monotonically increasing, therefore, C0(r) > 0 for r > r0. in Mq, Thus, to satisfy (1) and (7), C(r) must satisfy, 1 2 − 2 α q C0(r) 1. C0(r) > 0, r > r0; Rp(r) = 1 + dr = − C(r) C(r) 2 C(r) C(r) Z 2. lim = 1; r 2 p p →∞ r r0 = C(r) α C(r) + q2 + − (9) 2 2 2 2 − 3. C(r 0) = β = m + m q , q < m . q p − C(r) m + C(r) α C(r) + q2 I callp the foregoing thep Metric Conditions of Abrams + m ln − − , (MCA) for the point-charge. Abrams [6] obtained them by a q K p p different method — using (1) and the field equations directly.
In the absence of charge (7) must reduce to the general
K = const. Schwarzschild solution for the simple point-mass (2). The only functions that satisfy this requirement and MCA are, The valid relationship between and is, r Rp(r) 2 n n n Cn(r) = r r0 + β , as r r0 ,Rp(r) 0 , − → → h i so by (9), β = m + m2 q2, q2 < m2, − 2 2 +p r r0 C(r0) = m m q , n , r ( −) , → ⇒ ± − ∈ < 0 ∈ < − < q p where n and r0 are arbitrary. Therefore, the general solution K = m2 q2 . ± − for the point-charge is, 1 When q = 0, (9) must reducep to the Droste/Weyl [4, 5] α q2 α q2 − ds2 = 1 + dt2 1 + solution, so it requires, − √ C − − √ C × C C (15) 2 C0 2 2 2 2 C(r ) = m + m2 q2 . (10) dr C(dθ + sin θdϕ ) , 0 − × 4C − q p S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric 75 Volume 1 PROGRESS IN PHYSICS April, 2005
2 χ n n C (r) = r r + βn , lim , n 0 r r Rp → ∞ − → 0 h i which shows that Rp(r ) is a quasiregular singularity and β = m + m2 q2 , q2 < m2 , 0 − cannot be extended. + n p, r ( −) , Consider the Lagrangian, ∈ < 0 ∈ < − < 1 α q2 dt 2 r0 < r < . L = 1 + ∞ 2 − √C C dτ − " n n # When n = 1 and r0 = 0, Abrams’ [6] solution for the 2 1 2 point-charge results. 1 α q − d√Cn Equation (15) is regular r > r . There is no event 1 + (17) 0 − 2 − √C Cn dτ − horizon and therefore no Reissner-Nordstrom∀ black hole. " n # Furthermore, the Graves-Brill black hole and the Carter black 1 dθ 2 dϕ 2 hole are also invalid. C + sin2 θ . − 2 n dτ dτ By (15) the correct rendering of (3) is, " !# Restricting motion to the equatorial plane without loss of 2 2 1 2 α q 2 α q − 2 generality, the Euler-Lagrange equations from (17) are, ds = 1 + 2 dt 1 + 2 dr − r r − − r r − (16) 2 2 2 2 α q d √Cn α q dt 2 2 2 2 1 + 2 + 3 r (dθ + sin θdϕ ) , −√C Cn dτ 2Cn − 2 dτ − − n Cn ! 2 2 2 2 2 2 2 2 − √ q < m , m + m q < r < , α q α q d Cn (18) − ∞ 3 1 + − 2Cn − 2 − √C Cn dτ − p Cn ! n so Nordstrom’s assumption that C(0) = 0 is invalid. ijkm 2 The scalar curvature f = RijkmR for (1) with charge dϕ p Cn = 0 , included is, − dτ 2 p 8 6 m√C q2 + q4 α q2 dt − 1 + = k = const , (19) f = . − √C C dτ C4 n n dϕ Using (15) the curvature is, Cn = h = const . (20) dτ
1 2 2 μ ν Also, ds = gμν dx dx becomes, n n n 2 4 8 6 m r r0 + β q + q − − 2 " # α q2 dt f = h i 8 . 1 + n n − √C C dτ − r r + βn n n − 0 2 1 2 h i α q − d√Cn The curvature is always finite, even at r . No curvature 1 + (21) 0 − − √C C dτ − singularity can arise in the gravitational field of the point- n n charge. Furthermore, dϕ 2 Cn = 1 . 2 − dτ 8 6 mβ q2 + q4 f(r ) = − , It follows from these equations that the angular velocity 0 h 8 i β ω of a test particle is,
2 2 2 where β = m + m q . Thus, f(r0) is a scalar invariant 2 α q − 12 ω = = for the point-charge. When q = 0, f(r ) = , which is the 3 2 p 0 α4 2C 2 − Cn ! scalar curvature invariant for the simple point-mass. n (22) From (15) the circumference χ of a great circle is 2 given by, α q = 3 4 . n n − n n χ = 2π C(r) . 2 r r +βn r r +βn − 0 − 0 The proper radius is givenp by (12). Then the ratio χ >2π h i h i Rp Then, for finite r and, χ α q2 lim = 2π , lim ω = 3 4 , (23) r r r 2β − β →∞ Rp → 0 s
76 S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric April, 2005 PROGRESS IN PHYSICS Volume 1
2 2 2 2 where β = m + m q , q < m . Δ 2 2 − ds2 = dt a sin θdϕ Equation (22) is Kepler’s 3rd Law for the point-charge. ρ2 − − p (31) It obtains the finite limit given in (23), which is a scalar 2 2 sin θ 2 ρ invariant for the point-charge. When q = 0, equations (22) 2 2 2 2 2 2 r +a dϕ adt dr ρ dθ , and (23) reduce to those for the simple point-mass, − ρ − − Δ − L α a = , ρ2 = r2 + a2 cos2 θ , ω = 3 , m 2 s2Cn Δ = r2 rα + a2, 0 < r < , 1 − ∞ lim ω = . where L is the angular momentum. r r α√2 → 0 If a = 0, equation (31) reduces to Hilbert’s [7] solution In the case of a photon in circular orbit about the point- for the simple point-mass, charge, (21) yields, 1 2 α 2 α − 2 1 α q2 ds = 1 dt 1 dr ω2 = 1 + , (24) − r − − r − (32) Cn − √C Cn n r2(dθ2 + sin2 θdϕ2) , and (18) yields, − 2 0 < r < . 2 1 α q ∞ ω = 3 . (25) √C 2Cn − 2 However, according to the general formula (2) the correct n Cn ! range for r in (32) is, Equating the two, denoting the stable photon radial coor- dinate by rph, and solving for the curvature radius Cph = C(r0) < r < , = C (r ), gives (since when q =0, C = 0), ∞ n (ph) ph 6 p q where C(r0) = α. Therefore (32) should be, p 3α + 9α2p 32q2 C = C (r ) = − , (26) 1 ph n (ph) p 2 α 2 α − 2 p 4 ds = 1 dt 1 dr p q − r − − r − (33) which is a scalar invariant. In terms of coordinate radii, 2 2 2 2 1 r (dθ + sin θdϕ ) , n n − 2 2 3α + 9α 32q α < r < . r = − βn + r , (27) ∞ ph 4n − 0 p Equation (33) is the Droste/Weyl solution. Since the r that appears in (32) is the same r appearing which depends upon the values of n and r0. in (31) and (33), taking (4) into account, the correct general When q = 0 equations (26) and (27) reduce to the corres- form of (31) is, ponding equations for the simple point-mass, Δ 2 ds2 = dt a sin2 θdϕ 3α ρ2 − − Cn(rph) = , (28) 2 2 q sin θ 2 2 1 C + a dϕ adt (34) n n 2 3α n − ρ − − rph = α + r0 . (29) 2 − ρ2 C 2 0 dr2 ρ2dθ2 , The proper radius associated with (28) and (29) is, − Δ 4C − L α√3 1 + √3 a = , ρ2 = C + a2 cos2 θ , Rp(ph) = + α ln , (30) m 2 √2 ! Δ = C α√C + a2 , r < r < . which is a scalar invariant for the simple point-mass. Putting − 0 ∞ (26) into (12) gives the invariant proper radius for a stable When a = 0, (34) must reduce to (2). photon orbit about the point-charge. If C0 = 0 the structure of (34) is destroyed, since then g = 0 r > r B(r) = 0 in violation of (1). Therefore 11 ∀ 0 ⇒ 3 The vacuum field of the rotating point-mass C0 = 0. Equation (34) must have a global arrow for time, 6 whereupon g00(r0 = 0, so The Kerr solution, in Boyer-Lindquist coordinates and rela- tivistic units is, Δ(r ) = C(r ) α C(r ) + a2 = a2 sin2 θ . (35) 0 0 − 0 q S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric 77 Volume 1 PROGRESS IN PHYSICS April, 2005
Solving (35) for C(r0) gives, where rh is obtained from (31) by setting its Δ = 0, and r by setting its g = 0. Conventionally equations (42) are p b 00 β = C(r ) = m m2 a2 cos2 θ , (36) rather arbitrarily restricted to, 0 ± − q p r = m + m2 a2 , r = m + m2 a2 cos2 θ , (43) having used α = 2m. When a = 0 (36) must reduce to the va- h − b − lue for Schwarzschild’s [8] original solution, i. e. C(r ) = 0 p a2 < m2 . p = α = 2m, therefore the plus sign must be taken in (36). p Since the angular momentum increases the gravitational For (34), Δ 0 and so there is no static limit, since mass, and since there can be no angular momentum without by (41), ≥ 2 2 2 2 2 mass, a < m . Thus, there exists no spacetime for a > m . Cn(r0) = β , (44) To reduce to (2) equation (36) becomes, ⇒ Δ(r ) = β2 αβ + a2 . ⇒ 0 − β = C(r ) = m + m2 a2 cos2 , (37) Solving (41) i .e . 0 − q p 1 2 2 n n a < m . C (r) = r r +βn , (45) n − 0 Equation (34) must be asymptotically flat, so p h i gives the r-parameter location of a spacetime event,
C(r) 1 r 1 . (38) 1 n 2 2n n → ∞ ⇒ r r → r = Cn(r) β + r0 . (46) − 0 − h i Therefore, When a = 0, equation (46) reduces to r = α, as expected C(r) 0 lim = 1 . (39) for the non-rotating point-mass. r 2 →∞ r r From (46) it is concluded that there exists no spacetime − 0 drag effect for the rotating point-mass and no ergosphere. 2 Since C(r) behaves like r r , must make (34) The generalisation of equation (34) is then, − 0 singular only at r = r0, and C0(r) > 0 r > r0, C(r) is ∀ Δ 2 strictly monotonically increasing, so ds2 = dt a sin2 θdϕ ρ2 − − C0(r) > 0, r > r0 . (40) 2 sin θ 2 2 (47) 2 C + a dϕ adt Consequently, the conditions that C(r) must satisfy to − ρ − − render a solution to (34) are: ρ2 C 2 0 dr2 ρ2dθ2 , 1. C0(r) > 0, r > r0; − Δ 4C − C(r) 2 2. lim = 1; n n n + r 2 Cn(r) = r r0 +β , n , →∞ r r0 − ∈ < − h i 2 2 2 2 2 2 2 2 2 2 3. C(r ) = β= m + √m a cos θ, a < m . r ( −) , β = m+ m a cos θ , a < m , 0 − 0 ∈ <−< − I callp the foregoing the Metric Conditions of Abrams L p a = , ρ2 = C + a2 cos2 θ , (MCA) for the rotating point-mass. m n The only form admissible for C(r) in (34) that satisfies Δ = C α C + a2 , MCA and is reducible to (2) is, n − n 2 r0 < rp < . n n n ∞ Cn(r) = r r0 + β , (41) − Equation (47) is regular r > r0, and g00 = 0 only when h i r = r . There is no event horizon∀ and therefore no Kerr black 2 2 0 β = m + m2 a2 cos2 θ , a < m , hole. − By (47) the correct expression for the Kerr solution p + r ( −) n . 0 ∈ < − < ∈ < (31) is, Δ 2 Associated with (31) are the so-called “horizons” and 2 2 ds = 2 dt a sin θdϕ “static limits” given respectively by, ρ − − (48) 2 2 sin θ 2 ρ r2 +a2 dϕ adt dr2 ρ2dθ2 , r = m m2 a2 , r = m m2 a2 cos2 θ , (42) 2 h ± − b ± − − ρ − − Δ − p p 78 S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric April, 2005 PROGRESS IN PHYSICS Volume 1
L Δ = r2 rα + a2, a = , ρ2 = r2 + a2 cos2 θ , By (50) the correct expression for the Kerr-Newman − m solution (49) is, 2 2 2 2 2 a < m , m + m a cos θ < r < . Δ 2 − ∞ ds2 = dt a sin2 θdϕ When a = 0 in (48)p the Droste/Weyl solution (33) is ρ2 − − recovered. 2 2 sin θ 2 2 2 ρ 2 2 2 2 r + a dϕ adt dr ρ dθ , − ρ − − Δ − (51) 4 The vacuum field of the rotating point-charge L a= , ρ2 =r2+a2 cos2 θ, Δ=r2 rα+a2+q2, The Kerr-Newman solution is, in relativistic units, m −
2 2 2 2 2 2 2 2 Δ 2 2 q +a
S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric 79 Volume 1 PROGRESS IN PHYSICS April, 2005
two bodies of comparable mass. It is not even known if 12. Hawking S., Ellis G. F. R. The large scale structure of space- solutions to such configurations exist. One simply cannot talk time. Cambridge University Press, Cambridge, 1973. of black hole binaries or colliding black holes unless it can be shown, as pointed out by McVittie [11], that Einstein’s field equations admit of solutions for such configurations. Without such an existence theorem these ideas are without any theoretical basis. McVittie’s existence theorem however, does not exist, because the black hole does not exist in the formalism of General Relativity. It is also commonly claimed that the Michell-Laplace dark body is a kind of black hole or an anticipation of the black hole [10, 12]. This claim is utterly false as there always exists a class of observers who can see a Michell-Laplace dark body [11]: ipso facto, it is not a black hole. Consequently, there is no theoretical basis whatsoever for the existence of black holes. If such an object is ever detected then both Newton and Einstein would be invalidated.
Dedication
I dedicate this paper to the memory of Dr. Leonard S. Abrams: (27 Nov. 1924 — 28 Dec. 2001).
References
1. Crothers S. J. On the general solution to Einstein’s vacuum field and it implications for relativistic degeneracy. Progress in Physics, v. 1, 2005, 68–73. 2. Abrams L. S. Black holes: the legacy of Hilbert’s error. Can. J. Phys., v. 67, 1989, 919 (see also in arXiv: gr-qc/0102055). 3. Nordstrom G. K. Nederlandse Akad. Van Wetenschappen, Proceedings, v. 20, 1918, 1238. 4. Droste J. The field of a single centre in Einstein’s theory of gravitation, and the motion of a particle in that field. Ned. Acad. Wet., S. A., v. 19, 1917, 197 (see also in www.geocities.com/ theometria/Droste.pdf). 5. Weyl H. Zur Gravitationstheorie. Ann. d. Phys. (Leipzig), v. 54, 1917, 117. 6. Abrams L. S. The total space-time of a point charge and its consequences for black holes. Int. J. Theor. Phys., v. 35, 1996, 2661 (see also in arXiv: gr-qc/0102054). 7. Hilbert, D. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl., v. 53, 1917 (see also in arXiv: physics/0310104). 8. Schwarzschild K. On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 1916, 189 (see also in arXiv: physics/ 9905030). 9. Einstein A., Rosen, N. The particle problem in the General Theory of Relativity. Phys. Rev. v. 48, 1935, 73. 10. Misner C. W., Thorne K. S., Wheeler J. A. Gravitation. W. H. Freeman and Company, New York, 1973. 11. McVittie G. C. Laplace’s alleged “black hole”. The Observato- ry, v. 98, 1978, 272 (see also in www.geocities.com/theometria/ McVittie.pdf).
80 S J. Crothers. On the Ramifications of the Schwarzschild Space-Time Metric April, 2005 PROGRESS IN PHYSICS Volume 1
Experiments with Rotating Collimators Cutting out Pencil of α-Particles at Radioactive Decay of 239Pu Evidence Sharp Anisotropy of Space
Simon E. Shnoll∗, Ilia A. Rubinshtein†, Konstantin I. Zenchenko∗, Vladimir A. Shlekhtarev∗, Alexander V. Kaminsky, Alexander A. Konradov‡, Natalia V. Udaltsova∗
∗Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, 142290, Russia E-mail: [email protected]
†Skobeltsyn Inst. of Nuclear Physics, Moscow State Univ., Moscow, 119992, Russia ‡Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, Russia
As shown in our previous experiments fine structure of histograms of α-activity measurements serve as a sensitive tool for investigation of cosmo-physical influences. Particularly, the histograms structure is changed with the period equal to sidereal (1436 min) and solar (1440) day. It is similar with the high probability in different geographic points at the same local (longitude) time. More recently investigations were carried out with collimators, cutting out separate flows of total α-particles flying out at radioactive decay of 239Pu. These experiments revealed sharp dependence the histogram structure on the direction of α-particles flow. In the presented work measurements were made with collimators rotating in the plane of sky equator. It was shown that during rotation the shape of histograms changes with periods determined by number of revolution. These results correspond to the assumption that the histogram shapes are determined by a picture of the celestial sphere, and also by interposition of the Earth, the Sun and the Moon.
1 Introduction collimators, rotated counter-clockwise, west to east (i. e. in a direction of rotation of the Earth), as well as clockwise It has been earlier shown, that the fine structure of statistical (east to west). The description of these experiments is given distributions of measurement results of processes of various further. nature depends on cosmo-physical factors. The shape of corresponding histograms changes with the period equal to 2 Methods sidereal and solar day, i. e. 1436 and 1440 minutes [1, 2, 3, 4]. These periods disappeared at measurements of alpha- As well as earlier, the basic object of these of research was activity of 239Pu samples near the North Pole [5]. These a set of histograms constructed by results of measurements results corresponded to the assumption of association of the of alpha-activity of samples 239Pu. histogram shapes with a picture of the celestial sphere, and Experimental methods, the devices for alpha-radioactivity also with interposition of the Earth, the Sun and the Moon. measurements of 239Pu samples with collimators, and also However, at measurements at latitude 54◦ N (in Pushchi- construction of histograms and analysis of its shapes, are no), absence of the daily period [9] also was revealed when described in details in the earlier publications [2, 3, 8]. using collimators restricting a flow of the alpha particles of Measurements of number of events of radioactive decay were radioactive decay at the direction to the north celestial pole. completed by device designed by one of the authors (I. A. R.). This result meant, that the question is not about dependence In this device the semi-conductor detector (photo diode) on a picture of the celestial sphere above a place of measure- is placed after collimator, restricting a flow of the alpha ments, but about a direction of alpha particles flow. particles in a certain direction. Results of measurements, In experiments with two collimators, directed one to the consecutive numbers of events of the decay registered by East and another to the West, it was revealed, that histograms the detector in 1-second intervals, are stored in computer of the similar shape at measurements with west collimator archive. appear at 718 minutes (half of sidereal day) later then ones Depending on specific targets, a time sequence of1- registered with East collimator [9]. Therefore, as acquired, second measurements was summarized to consecutive values the space surrounding the Earth is highly anisotropic, and this of activity for 6, 15 or 60 seconds. Obtained time series anisotropy is connected basically to a picture of the celestial were separated into consecutive pieces of 60 numbers in sphere (sphere of distant stars). each. A histogram was built for each piece of 60 numbers. This suggestion has been confirmed in experiments with Histograms were smoothed using the method of moving
S. E. Shnoll et al. Experiments with Radioactive Decay of 239Pu Evidence Sharp Anisotropy of Space 81 Volume 1 PROGRESS IN PHYSICS April, 2005
averages for the greater convenience of a visual estimation 140 24 of similarity of their shapes (more details see in [8, 9]). 120 Comparison of histograms was performed using auxiliary computer program by Edwin Pozharski [8]. 100 A mechanical device designed by one of the authors 1 80 (V.A. Sh.) was used in experiments with rotation of collima- tors. In this device the measuring piece of equipment with 60 collimator was attached to the platform rotated in a plane of 40 Celestial Equator. 20 Number of similar histogram pairs 3 Results 0 6 12 18 24 Three revolutions of collimator counter-clockwise in a day. Time intervals, hours The diurnal period of increase in frequency of histograms Fig. 1: Frequency of similar 60-minute histograms against the time with similar shape means dependence of an observable pic- interval between histograms. Measurements of alpha-activity of a ture on rotation of the Earth. 239Pu sample by detector with motionless collimator directed to the The period of approximately 24 hours or with higher West, June 8 –30, 2004. resolution 1436 minutes is also observed at measurements using collimators restricting a flow of alpha particles in 60 a certain direction [9, 10]. Therefore, the fine structure of distribution of results of measurements depends on what site 50 of celestial sphere the flow of alpha particles is directed to. 40 Studies of shapes of histograms constructed by results of 30 measurements using rotated collimators testify to the benefit of this assumption. 20 The number of the “diurnal” cycles at clockwise rotation 10 should be one less then numbers of collimator revolutions 0 because of compensation of the Earth rotation. Number of similar histogram pairs 6 12 18 24 At May 28 through June 10, 2004, we have performed Time intervals, hours measurements of alpha-activity of a sample 239Pu at 3 col- Fig. 2: Frequency of similar 60-minute histograms against the time limator revolutions a day, and also, for the control, simul- interval between histograms. Measurements of alphaactivity of a taneous measurements with motionless collimator, directed 239Pu sample by detector with collimator, making three revolutions to the West. Results of these measurements are presented counter-clockwise (west to east) in a day. on Fig. 1– 4. At these figures a dependence of frequency histograms of the same shape on size of time interval between similar histograms is shown. revolutions counter-clockwise. At measurements with mo- Fig.1 shows results of comparison of 60-minute histo- tionless collimator (Fig. 3) there are two peaks — one cor- grams, constructed at measurements with motionless colli- responds to sidereal day (1436 minutes), the second, which mator. A typical dependence repeatedly obtained in earlier is less expressed, corresponds to solar day (1440 minutes). studies is visible at the Fig. 1: histograms of the same shape You can see at Fig. 4 that 6-hour period at measurements most likely appear at the nearest intervals of time (“effect of with three revolutions of collimator also has two components. a near zone”) and in one day (24 hours). The first 6-hour maximum has two joint peaks of 359 and Fig. 2 presents the result of comparison of 60-minute 360 minutes. The second 12-hour maximum has two peaks histograms constructed at measurements with collimator ro- of 718 and 720 minutes. The third maximum (18 hours) has tated 3 times a day counter-clockwise in a plane of celestial two peaks of 1077 and 1080 minutes. And the fourth one (24 equator. hours) has two peaks of 1436 and 1440 minutes. As you can see at the Fig. 2, at three revolutions of Results of these experiments confirm a conclusion ac- collimator counter-clockwise, the frequency of similar histo- cording to which a change in histogram shape is caused grams fluctuates with the period of 6 hours: peaks correspond by change in direction of alpha particles flow in relation to the intervals of 6, 12, 18 and 24 hours. to distant stars and the Sun (and other space objects). This 24-hour period at a higher resolution consists of two conclusion is supported also by results of experiments with components. It is visible by comparison of one-minute his- rotation of collimator clockwise. tograms shown at Fig. 3 for measurements with motionless In these experiments collimator made one revolution a collimator and at Fig. 4 for measurements at 3 collimator day clockwise, east to west, i. e. against daily rotation of the
82 S. E. Shnoll et al. Experiments with Radioactive Decay of 239Pu Evidence Sharp Anisotropy of Space April, 2005 PROGRESS IN PHYSICS Volume 1
Earth. As a result, the flow of alpha particles all the time was 140 1436 directed to the same point of celestial sphere. We expected 120 in this case disappearance the diurnal period of frequency of 100 similar histograms. This expectation was proved to be true. 80 1440 250 60 1436 200 1436 40 1440 1440 20 150 0 1434 1436 1438 1440 1442 1434 1436 1438 1440 1442 100 Time intervals, minutes Fig. 6: One-minute histograms. Left: control, motionless collimator. 50 Right: rotation 1 revolution clockwise (east to west).
Number of similar histogram pairs 0 On Fig. 5 and 6 one can see that in such experiments 1434 1436 1438 1440 1442 frequency of appearance of similar 60 minute and 1-minute Time intervals, minutes histograms does not depend on time. At the same time at Fig. 3: 24-hour period of frequency of similar histograms with the synchronous measurements with motionless collimator the one-minute resolution. Measurements of May 29 — June 1, 2004 usual dependence with the diurnal period and near zone effect by detector with motionless collimator directed to the West. is observed. 160
140 1077 1436 4 Discussion
120 720 1440 718 Results of measurements with rotated collimator confirm a 100 1080 359 conclusion about dependence of fine structure of statistical 80 360 distributions on a direction in space. This fine structure 60 is defined by a spectrum of amplitudes of fluctuations of
40 measured values. Presence of “peaks” and “hollows” at cor- responding histograms suggests presence of the primary, 20 allocated, “forbidden” and “permissible” values of ampli- 0 tudes of fluctuations in each given moment [4]. Thus, afine 356 359 362 716 719 722 725 1083 1437 1434 1440 1074 1077 1080 structure of statistical distributions presents a spectrum of the One-minute intervals permissible amplitudes of fluctuations, and dependence of it Fig. 4: Experiments with rotated collimators. Frequency of similar on a direction in space shows sharp anisotropy of space. 1-minute histograms by time interval between them. Three revo- It is necessary to emphasize, that the question is not lutions a day counter-clockwise. Two components of the 6-hour about influence on the subject of measurement (in this case period: sidereal and solar. on radioactive decay). With accuracy of traditional statistical 140 criteria, overall characteristics of distribution of radioactive 24 decay measurements compliant with Poisson distribution [3]. 120 Only the shape of histogram constructed for small sample
100 size varies regularly. This regularity emerges in precise si- 1 dereal and solar periods of increase of frequency of similar 80 histograms. As shown above, the shape of histograms constructed by 60 1 24 results of measurements of alpha-activity of samples 239Pu, 40 varies with the period determined by number of revolutions in relation to celestial sphere and the Sun. In experiments with 20 collimator, which made three revolutions counter-clockwise,
0 the “diurnal” period was equal to 6 hours (three revolutions 6 12 18 24 6 12 18 24 of collimator and one revolution of the Earth was observed Time intervals, hours — in total 4 revolutions in relation to celestial sphere and the Fig. 5: 60-minutes histograms. Left: 1 revolution clockwise. Right: Sun give the period equal 24/4 = 6 hours). control, motionless collimator. The result obtained in experiments with one revolution of
S. E. Shnoll et al. Experiments with Radioactive Decay of 239Pu Evidence Sharp Anisotropy of Space 83 Volume 1 PROGRESS IN PHYSICS April, 2005
collimator clockwise is not less important. The Earth rotation scopic fluctuations” on cosmophysical factors. Spatial aniso- is compensated and a flow of alpha particles is directed tropy. Biophysics, 2004, v. 49(1), 129–139. all the time to the same point of celestial sphere. In these 7. Shnoll S. E., Zenchenko K. I., Berulis I. I., Udaltsova N. V. and experiments the diurnal period was not observed at all. Rubinstein I. A. Fine structure of histograms of alpha-activity The obtained results, though very clear ones, cause nat- measurements depends on direction of alpha particles flow ural bewilderment. and the Earth rotation: experiments with collimators. arXiv: Really, it is completely not obvious, by virtue of what physics/0412007. reasons the spectrum of amplitudes of fluctuations of number 8. Shnoll S.E., Kolombet V.A., Zenchenko T.A., Pozharskii E.V., of alpha particles, may depend on a direction of their flow Zvereva I.M. and Konradov A.A. Cosmophysical origin in relation to celestial sphere and the Sun. The explanation of “macroscopic fluctations”. Biophysics, 1998, v. 43(5), of these phenomena probably demands essential change in 909–915. general physical conceptions. 9. Fedorov M. V., Belousov L. V., Voeikov V.L., Zenchenko T. A., In such situation a dominant problem is to validate a Zenchenko K. I., Pozharskii E. V., Konradov A. A. and reliability of the discussed phenomena. In aggregate of per- Shnoll S. E. Synchronous changes in dark current fluctuations in two separate photomultipliers in relation to Earth rotation. formed studies, we believe this task was completed . Astrophysics and Space Science, 2003, v. 283, 3–10.
5 Acknowledgements
S. E. Shnoll is indebted to M. N. Kondrashova and L. A. Blu- menfeld for mutual understanding. The authors are grateful to our colleagues, Т. А. Zen- chenko, D. P.Kharakoz, B. M. Vladimirsky, B. V.Komberg, V.K. Lyapidevskii for collaboration and for valuable discus- sions. The vivid interest of V.P.Tikhonov to the problem stud- ied and his generous financial support were essential.
References
1. Shnoll S. E., Kolombet V.A., Pozharski E. V., Zenchenko T. A., Zvereva I. M. and Konradov A. A. Realization of discrete states during fluctuations in macroscopic processes. Physics-Uspehi, 1998, v. 162(10), 1129–1140. 2. Shnoll S. E., Pozharski E. V., Zenchenko T. A., Kolombet V.A., Zvereva I. M. and Konradov A. A. Fine structure of distributions in measurements of different processes as affected by geophysical and cosmophysical factors. Phys. and Chem. Earth A: Solid Earth and Geod., 1999, v. 24(8), 711–714. 3. Shnoll S. E., Zenchenko T. A., Zenchenko K. I., Pozhar- ski E. V., Kolombet V.A. and Konradov A. A. Regular variation of the fine structure of statistical distributions as a consequence of cosmophysical agents. Physics-Uspehi, 2000, v. 43(2), 205–209. 4. Shnoll S. E. Discrete distribution patterns: arithmetic and cosmophysical origins of their macroscopic fluctuations. Biophysics, 2001, v. 46(5), 733–741. 5. Shnoll S. E., Rubinstein I. A., Zenchenko K. I., Zenchen- ko T. A., Konradov A. A., Shapovalov S. N., Makarevich A. V., Gorshkov E. S. and Troshichev O. A. Dependence of “macroscopic fluctuations” on geographic coordinates (by materials of Arctic and Antarctic expeditions). Biophysics, 2003, v. 48(5), 1123–1131. 6. Shnoll S. E., Zenchenko K. I., Berulis I. I., Udaltsova N. V., Zhirkov S. S. and Rubinstein I. A. Dependence of “macro-
84 S. E. Shnoll et al. Experiments with Radioactive Decay of 239Pu Evidence Sharp Anisotropy of Space April, 2005 PROGRESS IN PHYSICS Volume 1
There Is No Speed Barrier for a Wave Phase Nor for Entangled Particles
Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 College Road, Gallup, NM 87301, USA E-mail: [email protected]; [email protected]
In this short paper, as an extension and consequence of Einstein-Podolski-Rosen paradox and Bell’s inequality, one promotes the hypothesis (it has been called the Smarandache Hypothesis [1, 2, 3]) that: There is no speed barrier in the Universe and one can construct arbitrary speeds, and also one asks if it is possible to have an infinite speed (instantaneous transmission)? Future research: to study the composition of faster-than-light velocities and what happens with the laws of physics at faster-than- light velocities?
This is the new version of an early article. That early version, sert that the light speed is not a speed barrier in the based on a 1972 paper [4], was presented at the Universidad Universe. de Blumenau, Brazil, May–June 1993, in the Conference Such results were also obtained by: Nicolas Gisin at on “Paradoxism in Literature and Science”; and at the Uni- the University of Geneva, Switzerland, who successfully versity of Kishinev, in December 1994. See that early ver- teleported quantum bits, or qubits, between two labs over sion in [5]. 2 km of coiled cable. But the actual distance between the two labs was about 55 m; researchers from the University of Vienna and the Austrian Academy of Science (Rupert Ursin 1 Introduction et al. have carried out successful teleportation with particles What is new in science (physics)? of light over a distance of 600 m across the River Danube in According to researchers from the common group of the Austria); researchers from Australia National University and University of Innsbruck in Austria and US National Institute many others [6, 7, 8]. of Standards and Technology (starting from December 1997, Rainer Blatt, David Wineland et al.): 2 Scientific hypothesis Photon is a bit of light, the quantum of electromagnetic • radiation (quantum is the smallest amount of energy We even promote the hypothesis that: that a system can gain or lose); There is no speed barrier in the Universe, which would theoretically be proved by increasing, in the previous Polarization refers to the direction and characteristics • example, the distance between particles A and B as of the light wave vibration; much as the Universe allows it, and then measuring If one uses the entanglement phenomenon, in order to particle A. • transfer the polarization between two photons, then: It has been called the Smarandache Hypotesis [1, 2, 3]. whatever happens to one is the opposite of what hap- pens to the other; hence, their polarizations are oppos- 3 An open question now ite of each other; In quantum mechanics, objects such as subatomic par- • If the space is infinite, is the maximum speed infinite? ticles do not have specific, fixed characteristic at any “This Smarandache hypothesis is controversially inter- given instant in time until they are measured; preted by scientists. Some say that it violates the theory of Suppose a certain physical process produces a pair relativity and the principle of causality, others support the • of entangled particles A and B (having opposite or ideas that this hypothesis works for particles with no mass complementary characteristics), which fly off into spa- or imaginary mass, in non-locality, through tunneling effect, ce in the opposite direction and, when they are billions or in other (extra-) dimension(s).” Kamla John, [9]. of miles apart, one measures particle A; because B is Scott Owens’ answer [10] to Hans Gunter in an e-mail the opposite, the act of measuring A instantaneously from January 22, 2001 (the last one forwarded it to the tells B what to be; therefore those instructions would author): “It appears that the only things the Smarandache somehow have to travel between A and B faster than hypothesis can be applied to are entities that do not have real the speed of light; hence, one can extend the Einstein- mass or energy or information. The best example I can come Podolsky-Rosen paradox and Bell’s inequality and as- up with is the difference between the wavefront velocity of
F. Smarandache. There Is No Speed Barrier for a Wave Phase Nor for Entangled Particles 85 Volume 1 PROGRESS IN PHYSICS April, 2005
a photon and the phase velocity. It is common for the phase 9. Kamla J. Private communications. 2001. velocity to exceed the wavefront velocity c, but that does 10. Owens S. Private communications. 2001. not mean that any real energy is traveling faster than c. So, while it is possible to construct arbitrary speeds from zero in infinite, the superluminal speeds can only apply to purely imaginary entities or components.” Would be possible to accelerate a photon (or another particle traveling at, say, 0.99c and thus to get speed greater than c (where c is the speed of light)?
4 Future possible research
It would be interesting to study the composition of two velocities v and u in the cases when: v < c and u = c; v = c and u = c; v > c and u = c; v > c and u > c; v < c and u = ; ∞ v = c and u = ; ∞ v > c and u = ; ∞ v = and u = . ∞ ∞ What happens with the laws of physics in each of these cases?
References
1. Motta L. Smarandache Hypothesis: Evidences, Implications, and Applications. Proceedings of the Second Internation- al Conference on Smarandache Type Notions In Mathem- atics and Quantum Physics, University of Craiova, Craiova, Romania, December 21–24, 2000 (see the e-print version in the web site at York University, Canada, http://at.yorku.ca/cgi- bin/amca/caft-03). 2. Motta L. and Niculescu G., editors. Proceedings of the Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics, American Research Press, 2000. 3. Weisstein E. W. Smarandache Hypothesis. The Encyclopedia of Physics, Wolfram Research (http://scienceworld.wolfram. com/physics/SmarandacheHypothesis.htm). 4. Smarandache F. Collected Papers. Vol.III, Abaddaba Publ. Hse., Oradea, Romania, 2000, 158. 5. Smarandache F. There is no speed barrier in the Universe. Bulletin of Pure and Applied Sciences, Delhi, India, v. 17D (Physics), No. 1, 1998, 61. 6. Rincon P. Teleportation breakthrough made. BBC News On- line, 2004/06/16. 7. Rincon P. Teleportation goes long distance. BBC News Online, 2004/08/18. 8. Whitehouse D. Australian teleport breakthrough. BBC News Online, 2002/06/17.
86 F. Smarandache. There Is No Speed Barrier for a Wave Phase Nor for Entangled Particles Progress in Physics is a quarterly issue scientific journal, registered with the Library of Congress (DC). This is a journal for scientific publications on advanced studies in theoretical and experimental physics, including related themes from mathematics. Electronic version of this journal: http://www.geocities.com/ptep_online
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